Ответ:
![-2,5, \quad -2, \quad 0,5, \quad 1; -2,5, \quad -2, \quad 0,5, \quad 1;](https://tex.z-dn.net/?f=-2%2C5%2C%20%5Cquad%20-2%2C%20%5Cquad%200%2C5%2C%20%5Cquad%201%3B)
Объяснение:
![(2x^{2}+3x-1)^{2}-10x^{2}-15x+9=0; (2x^{2}+3x-1)^{2}-10x^{2}-15x+9=0;](https://tex.z-dn.net/?f=%282x%5E%7B2%7D%2B3x-1%29%5E%7B2%7D-10x%5E%7B2%7D-15x%2B9%3D0%3B)
![(2x^{2}+3x-1)(2x^{2}+3x-1)-10x^{2}-15x+9=0; (2x^{2}+3x-1)(2x^{2}+3x-1)-10x^{2}-15x+9=0;](https://tex.z-dn.net/?f=%282x%5E%7B2%7D%2B3x-1%29%282x%5E%7B2%7D%2B3x-1%29-10x%5E%7B2%7D-15x%2B9%3D0%3B)
![4x^{4}+6x^{3}-2x^{2}+6x^{3}+9x^{2}-3x-2x^{2}-3x+1-10x^{2}-15x+9=0; 4x^{4}+6x^{3}-2x^{2}+6x^{3}+9x^{2}-3x-2x^{2}-3x+1-10x^{2}-15x+9=0;](https://tex.z-dn.net/?f=4x%5E%7B4%7D%2B6x%5E%7B3%7D-2x%5E%7B2%7D%2B6x%5E%7B3%7D%2B9x%5E%7B2%7D-3x-2x%5E%7B2%7D-3x%2B1-10x%5E%7B2%7D-15x%2B9%3D0%3B)
![4x^{4}+6x^{3}+6x^{3}-2x^{2}+9x^{2}-2x^{2}-10x^{2}-3x-3x-15x+1+9=0; 4x^{4}+6x^{3}+6x^{3}-2x^{2}+9x^{2}-2x^{2}-10x^{2}-3x-3x-15x+1+9=0;](https://tex.z-dn.net/?f=4x%5E%7B4%7D%2B6x%5E%7B3%7D%2B6x%5E%7B3%7D-2x%5E%7B2%7D%2B9x%5E%7B2%7D-2x%5E%7B2%7D-10x%5E%7B2%7D-3x-3x-15x%2B1%2B9%3D0%3B)
![4x^{4}+12x^{3}-5x^{2}-21x+10=0; 4x^{4}+12x^{3}-5x^{2}-21x+10=0;](https://tex.z-dn.net/?f=4x%5E%7B4%7D%2B12x%5E%7B3%7D-5x%5E%7B2%7D-21x%2B10%3D0%3B)
Делителями числа 10 являются:
![\pm 1, \quad \pm 2, \quad \pm 5, \quad \pm 10; \pm 1, \quad \pm 2, \quad \pm 5, \quad \pm 10;](https://tex.z-dn.net/?f=%5Cpm%201%2C%20%5Cquad%20%5Cpm%202%2C%20%5Cquad%20%5Cpm%205%2C%20%5Cquad%20%5Cpm%2010%3B)
![x=1 \Rightarrow 4+12-5-21+10=16-5-21+10=11-21+10=-10+10=0; x=1 \Rightarrow 4+12-5-21+10=16-5-21+10=11-21+10=-10+10=0;](https://tex.z-dn.net/?f=x%3D1%20%5CRightarrow%204%2B12-5-21%2B10%3D16-5-21%2B10%3D11-21%2B10%3D-10%2B10%3D0%3B)
Единица обращает уравнение в верное равенство. Разделим исходный многочлен на "х–1":
![\frac{4x^{4}+12x^{3}}{x-1}=\frac{4x^{4}-4x^{3}+16x^{3}}{x-1}=\frac{4x^{3}(x-1)+16x^{3}}{x-1}=4x^{3}+\frac{16x^{3}}{x-1}; \frac{4x^{4}+12x^{3}}{x-1}=\frac{4x^{4}-4x^{3}+16x^{3}}{x-1}=\frac{4x^{3}(x-1)+16x^{3}}{x-1}=4x^{3}+\frac{16x^{3}}{x-1};](https://tex.z-dn.net/?f=%5Cfrac%7B4x%5E%7B4%7D%2B12x%5E%7B3%7D%7D%7Bx-1%7D%3D%5Cfrac%7B4x%5E%7B4%7D-4x%5E%7B3%7D%2B16x%5E%7B3%7D%7D%7Bx-1%7D%3D%5Cfrac%7B4x%5E%7B3%7D%28x-1%29%2B16x%5E%7B3%7D%7D%7Bx-1%7D%3D4x%5E%7B3%7D%2B%5Cfrac%7B16x%5E%7B3%7D%7D%7Bx-1%7D%3B)
![\frac{16x^{3}-5x^{2}}{x-1}=\frac{16x^{3}-16x^{2}+11x^{2}}{x-1}=\frac{16x^{2}(x-1)+11x^{2}}{x-1}=16x^{2}+\frac{11x^{2}}{x-1}; \frac{16x^{3}-5x^{2}}{x-1}=\frac{16x^{3}-16x^{2}+11x^{2}}{x-1}=\frac{16x^{2}(x-1)+11x^{2}}{x-1}=16x^{2}+\frac{11x^{2}}{x-1};](https://tex.z-dn.net/?f=%5Cfrac%7B16x%5E%7B3%7D-5x%5E%7B2%7D%7D%7Bx-1%7D%3D%5Cfrac%7B16x%5E%7B3%7D-16x%5E%7B2%7D%2B11x%5E%7B2%7D%7D%7Bx-1%7D%3D%5Cfrac%7B16x%5E%7B2%7D%28x-1%29%2B11x%5E%7B2%7D%7D%7Bx-1%7D%3D16x%5E%7B2%7D%2B%5Cfrac%7B11x%5E%7B2%7D%7D%7Bx-1%7D%3B)
![\frac{11x^{2}-21x}{x-1}=\frac{11x^{2}-11x-10x}{x-1}=\frac{11x(x-1)-10x}{x-1}=11x+\frac{-10x}{x-1}; \frac{11x^{2}-21x}{x-1}=\frac{11x^{2}-11x-10x}{x-1}=\frac{11x(x-1)-10x}{x-1}=11x+\frac{-10x}{x-1};](https://tex.z-dn.net/?f=%5Cfrac%7B11x%5E%7B2%7D-21x%7D%7Bx-1%7D%3D%5Cfrac%7B11x%5E%7B2%7D-11x-10x%7D%7Bx-1%7D%3D%5Cfrac%7B11x%28x-1%29-10x%7D%7Bx-1%7D%3D11x%2B%5Cfrac%7B-10x%7D%7Bx-1%7D%3B)
![\frac{-10x+10}{x-1}=\frac{-10(x-1)}{x-1}=-10; \frac{-10x+10}{x-1}=\frac{-10(x-1)}{x-1}=-10;](https://tex.z-dn.net/?f=%5Cfrac%7B-10x%2B10%7D%7Bx-1%7D%3D%5Cfrac%7B-10%28x-1%29%7D%7Bx-1%7D%3D-10%3B)
![(x-1)(4x^{3}+16x^{2}+11x-10)=0; (x-1)(4x^{3}+16x^{2}+11x-10)=0;](https://tex.z-dn.net/?f=%28x-1%29%284x%5E%7B3%7D%2B16x%5E%7B2%7D%2B11x-10%29%3D0%3B)
![4x^{3}+16x^{2}+11x-10=0; 4x^{3}+16x^{2}+11x-10=0;](https://tex.z-dn.net/?f=4x%5E%7B3%7D%2B16x%5E%7B2%7D%2B11x-10%3D0%3B)
Делителями числа 10 являются:
![\pm 1, \quad \pm 2, \quad \pm 5, \quad \pm 10; \pm 1, \quad \pm 2, \quad \pm 5, \quad \pm 10;](https://tex.z-dn.net/?f=%5Cpm%201%2C%20%5Cquad%20%5Cpm%202%2C%20%5Cquad%20%5Cpm%205%2C%20%5Cquad%20%5Cpm%2010%3B)
![x=1 \Rightarrow 4+16+11-10=20+1=21 \neq 0; x=1 \Rightarrow 4+16+11-10=20+1=21 \neq 0;](https://tex.z-dn.net/?f=x%3D1%20%5CRightarrow%204%2B16%2B11-10%3D20%2B1%3D21%20%5Cneq%200%3B)
![x=-1 \Rightarrow -4+16-11-10=12-21=-9 \neq 0; x=-1 \Rightarrow -4+16-11-10=12-21=-9 \neq 0;](https://tex.z-dn.net/?f=x%3D-1%20%5CRightarrow%20-4%2B16-11-10%3D12-21%3D-9%20%5Cneq%200%3B)
![x=2 \Rightarrow 4 \cdot 8+16 \cdot 4+11 \cdot 2-10=32+64+22-10=118-10=108 \neq 0; x=2 \Rightarrow 4 \cdot 8+16 \cdot 4+11 \cdot 2-10=32+64+22-10=118-10=108 \neq 0;](https://tex.z-dn.net/?f=x%3D2%20%5CRightarrow%204%20%5Ccdot%208%2B16%20%5Ccdot%204%2B11%20%5Ccdot%202-10%3D32%2B64%2B22-10%3D118-10%3D108%20%5Cneq%200%3B)
![x=-2 \Rightarrow 4 \cdot (-8)+16 \cdot 4+11 \cdot (-2)-10=-32+64-22-10=0; x=-2 \Rightarrow 4 \cdot (-8)+16 \cdot 4+11 \cdot (-2)-10=-32+64-22-10=0;](https://tex.z-dn.net/?f=x%3D-2%20%5CRightarrow%204%20%5Ccdot%20%28-8%29%2B16%20%5Ccdot%204%2B11%20%5Ccdot%20%28-2%29-10%3D-32%2B64-22-10%3D0%3B)
Число –2 обращает второе уравнение в верное равенство. Разделим второй многочлен на "х+2":
![\frac{4x^{3}+16x^{2}}{x+2}=\frac{4x^{3}+8x^{2}+8x^{2}}{x+2}=\frac{4x^{2}(x+2)+8x^{2}}{x+2}=4x^{2}+\frac{8x^{2}}{x+2}; \frac{4x^{3}+16x^{2}}{x+2}=\frac{4x^{3}+8x^{2}+8x^{2}}{x+2}=\frac{4x^{2}(x+2)+8x^{2}}{x+2}=4x^{2}+\frac{8x^{2}}{x+2};](https://tex.z-dn.net/?f=%5Cfrac%7B4x%5E%7B3%7D%2B16x%5E%7B2%7D%7D%7Bx%2B2%7D%3D%5Cfrac%7B4x%5E%7B3%7D%2B8x%5E%7B2%7D%2B8x%5E%7B2%7D%7D%7Bx%2B2%7D%3D%5Cfrac%7B4x%5E%7B2%7D%28x%2B2%29%2B8x%5E%7B2%7D%7D%7Bx%2B2%7D%3D4x%5E%7B2%7D%2B%5Cfrac%7B8x%5E%7B2%7D%7D%7Bx%2B2%7D%3B)
![\frac{8x^{2}+11x}{x+2}=\frac{8x^{2}+16x-5x}{x+2}=\frac{8x(x+2)-5x}{x+2}=8x+\frac{-5x}{x+2}; \frac{8x^{2}+11x}{x+2}=\frac{8x^{2}+16x-5x}{x+2}=\frac{8x(x+2)-5x}{x+2}=8x+\frac{-5x}{x+2};](https://tex.z-dn.net/?f=%5Cfrac%7B8x%5E%7B2%7D%2B11x%7D%7Bx%2B2%7D%3D%5Cfrac%7B8x%5E%7B2%7D%2B16x-5x%7D%7Bx%2B2%7D%3D%5Cfrac%7B8x%28x%2B2%29-5x%7D%7Bx%2B2%7D%3D8x%2B%5Cfrac%7B-5x%7D%7Bx%2B2%7D%3B)
![\frac{-5x-10}{x+2}=\frac{-5(x+2)}{x+2}=-5; \frac{-5x-10}{x+2}=\frac{-5(x+2)}{x+2}=-5;](https://tex.z-dn.net/?f=%5Cfrac%7B-5x-10%7D%7Bx%2B2%7D%3D%5Cfrac%7B-5%28x%2B2%29%7D%7Bx%2B2%7D%3D-5%3B)
![(x-1)(x+2)(4x^{2}+8x-5)=0; (x-1)(x+2)(4x^{2}+8x-5)=0;](https://tex.z-dn.net/?f=%28x-1%29%28x%2B2%29%284x%5E%7B2%7D%2B8x-5%29%3D0%3B)
![4x^{2}+8x-5=0; 4x^{2}+8x-5=0;](https://tex.z-dn.net/?f=4x%5E%7B2%7D%2B8x-5%3D0%3B)
![D=b^{2}-4ac; D=b^{2}-4ac;](https://tex.z-dn.net/?f=D%3Db%5E%7B2%7D-4ac%3B)
![D=8^{2}-4 \cdot 4 \cdot (-5)=64+16 \cdot 5=64+80=144; D=8^{2}-4 \cdot 4 \cdot (-5)=64+16 \cdot 5=64+80=144;](https://tex.z-dn.net/?f=D%3D8%5E%7B2%7D-4%20%5Ccdot%204%20%5Ccdot%20%28-5%29%3D64%2B16%20%5Ccdot%205%3D64%2B80%3D144%3B)
![x_{1}=\frac{-b+\sqrt{D}}{2a}; x_{1}=\frac{-b+\sqrt{D}}{2a};](https://tex.z-dn.net/?f=x_%7B1%7D%3D%5Cfrac%7B-b%2B%5Csqrt%7BD%7D%7D%7B2a%7D%3B)
![x_{1}=\frac{-8+\sqrt{144}}{2 \cdot 4}=\frac{-8+12}{8}=\frac{4}{8}=0,5; x_{1}=\frac{-8+\sqrt{144}}{2 \cdot 4}=\frac{-8+12}{8}=\frac{4}{8}=0,5;](https://tex.z-dn.net/?f=x_%7B1%7D%3D%5Cfrac%7B-8%2B%5Csqrt%7B144%7D%7D%7B2%20%5Ccdot%204%7D%3D%5Cfrac%7B-8%2B12%7D%7B8%7D%3D%5Cfrac%7B4%7D%7B8%7D%3D0%2C5%3B)
![x_{2}=\frac{-b-\sqrt{D}}{2a}; x_{2}=\frac{-b-\sqrt{D}}{2a};](https://tex.z-dn.net/?f=x_%7B2%7D%3D%5Cfrac%7B-b-%5Csqrt%7BD%7D%7D%7B2a%7D%3B)
![x_{2}=\frac{-8-\sqrt{144}}{2 \cdot 4}=\frac{-8-12}{8}=\frac{-20}{8}=-2,5; x_{2}=\frac{-8-\sqrt{144}}{2 \cdot 4}=\frac{-8-12}{8}=\frac{-20}{8}=-2,5;](https://tex.z-dn.net/?f=x_%7B2%7D%3D%5Cfrac%7B-8-%5Csqrt%7B144%7D%7D%7B2%20%5Ccdot%204%7D%3D%5Cfrac%7B-8-12%7D%7B8%7D%3D%5Cfrac%7B-20%7D%7B8%7D%3D-2%2C5%3B)
Итого, корни уравнения:
![-2,5, \quad -2, \quad 0,5, \quad 1; -2,5, \quad -2, \quad 0,5, \quad 1;](https://tex.z-dn.net/?f=-2%2C5%2C%20%5Cquad%20-2%2C%20%5Cquad%200%2C5%2C%20%5Cquad%201%3B)
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