Verified answer

[tex]\displaystyle\bf\\1)\\\\x^{2} (x-2)-18x(x-2)+81(x-2)=(x-2)(x^{2} -18x+81)=\\\\=(x-2)(x-9)^{2} =(x-2)(x-9)(x-9)\\\\2)\\\\4x(y^{2} -9)+4x^{2} (y^{2} -9)-9+y^{2} =\\\\=4x(y^{2} -9)+4x^{2} (y^{2} -9)+(y^{2}-9) =(y^{2}-9)(4x^{2} +4x+1)=\\\\=(y-3)(y+3)(2x+1)^{2} =(y-3)(y+3)(2x+1)(2x+1)\\\\3)\\\\b^{2} (a+1)-a^{2}(b+1) =ab^{2} +b^{2}-a^{2}b-a^{2} =\\\\=(ab^{2} -a^{2} b)+(b^{2} -a^{2} )=ab\cdot(b-a)+(b-a)(b+a)=\\\\=(b-a)(ab+a+b)\\\\4)\\\\\\(a-b)(b^{2} - c^{2})-(b-c) (a^{2} -b^{2} )=[/tex]

[tex]\displaystyle\bf\\=(a-b)(b-c)(b+c)-(b-c)(a-b)(a+b)=\\\\=(a-b)(b-c)(b+c-a-b)=(a-b)(b-c)(c-a)\\\\\\1)\\\\x^{2} (x+4)-20x(x+4)+100(x+4)=\\\\=(x+4)(x^{2}-20x+100)=(x+4)(x -10)^{2} =\\\\=(x+4)(x-10)(x-10)\\\\2)\\\\a^{2}-36-2a(36-a^{2} )-a^{2} (36-a^{2} )=\\\\=(a^{2}-36)+2a(a^{2} -36)+a^{2} (a^{2} -36 )=\\\\=(a^{2} -36)(a^{2} +2a+1)=(a-6)(a+6)(a+1)^{2} =\\\\=(a-6)(a+6)(a+1)(a+1)\\\\3)\\\\a^{2} (b-1)-b^{2} (a-1)=a^{2} b-a^{2} -ab^{2} +b^{2} =[/tex]

[tex]\displaystyle\bf\\=(a^{2} b-ab^{2} )-(a^{2} -b^{2} )=ab(a-b)-(a-b)(a+b)=\\\\=(a-b)(ab-a-b)\\\\4)\\\\(m-n)(n^{3} -p^{3})-(n-p)(m^{3} -n^{3} )=\\\\=(m-n)(n-p)(n^{2} +np+p^{2} )-(n-p)(m-n)(m^{2}+mn+n^{2} )=\\\\=(m-n)(n-p)(n^{2} +np+p^{2} -m^{2}-mn-n^{2} )=\\\\=(m-n)(n-p)(np+p^{2} -m^{2}-mn )=\\\\=(m-n)(n-p)(p-m)(p+m+n)[/tex]