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August 2022 1 12 Report

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$1) \: {x}^{3} +3 x {}^{2} - x - 3 =( {x}^{3} +3 x {}^{2} ) + ( - x - 3) = {x}^{2} (x + 3) - (x + 3) = (x + 3)( {x}^{2} - 1) = (x + 3)(x - 1)(x + 1)$

$2) \: {x}^{3} + {x}^{2} - 4x - 4 = ( {x}^{3} + {x}^{2}) + ( - 4x - 4) = {x}^{2} (x + 1) - 4(x + 1) = (x + 1)(x {}^{2} - 4) = (x + 1)(x - 2)(x + 2)$

$3) \: b {}^{2} a + b {}^{2} - {a}^{3} - {a}^{2} =( b {}^{2} a + b {}^{2} ) + ( - {a}^{3} - {a}^{2}) = b {}^{2} (a + 1) - {a}^{2} (a + 1) = (a + 1)(b {}^{2} - a {}^{2} ) = (a + 1)(b - a)(b + a)$

$4) \: {x}^{3} - 4 {x}^{2} - x + 4 = ({x}^{3} - 4 {x}^{2}) + ( - x + 4) = {x}^{2} (x - 4) - (x - 4) = (x - 4)( {x}^{2} - 1) = (x - 4)(x + 1)(x - 1)$

$5) \: {x}^{3} + 6x {}^{2} - x - 6 =( {x}^{3} + 6x {}^{2} ) + ( - x - 6) = {x}^{2} (x + 6) - (x + 6) = (x + 6)( {x}^{2} - 1) = (x + 6)(x - 1)(x + 1)$

$6) \:2a + 2b + {a}^{2} + ab = (2a + 2b) + ({a}^{2} + ab) = 2(a + b) + a(a + b) = (a + b)(2 + a)$

$7) \: {m}^{2} + mn - m - mq - nq + q =( {m}^{2} + mn - m) + ( - mq - nq + q) = m(m + n - 1) - q(m + n - 1) = (m + n - 1)(m - q)$

$8) \: 4a {}^{2} - b {}^{2} + 2a - b = (4a {}^{2} - b {}^{2}) + (2a - b ) = (2a - b)(2a + b) + (2a - b) = (2a - b)(2a + b + 1) =$

$9) \: 2xy - 3ay + 2x {}^{2} - 3ax = (2xy + 2x {}^{2} ) + ( - 3ay - 3ax) = 2x(y + x) - 3a(y + x) = (y + x)(2x - 3a)$

$10) \: xy + {a}^{2} - ax - ay = (xy - ax) + ({a}^{2} - ay) = x(y - a) - a( - a + y) = (y - a)(x - a)$

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