Question

Factorize the following polynomial 1)(y+2) (y-3) (y+8) (y+3) + 56

Answer 1

${y (y + 3) + 2(y + 3)} \: {y(y + 8) - 3(y + 8)} \: + 56 \\(y {}^{2} + 3y + 2y + 6) \: (y {}^{2} + 8y - 3y - 24) + 56\\(y {}^{2} + 5y + 6) \: (y { }^{2} + 5y - 24) + 56$

Let y²+5y=x

$(x + 6) \: (x - 24) + 56 \\ \\ x(x - 24) + 6(x - 24) + 56 \\ \\ x {}^{2} - 24x + 6x - 144 + 56 \\ \\ x {}^{2} - 18x - 144 + 56 \\ \\ x {}^{2} - 18x - 88 \\ \\ x {}^{2} - 22x - 4x - 88 \\ \\ x(x - 22) + 4(x - 22) \\ \\ (x - 22) \: (x + 4) \\ \\ \\ substituting \: the \: value \: of \: x \: with \: y {}^{2} + 5y \\ \\ \\ (y {}^{2} + 5y - 22) \: (y {}^{2} + 5y + 4) \\ \\ (y {}^{2} + 5y - 22) \: (y {}^{2} + 4y + y + 4) \\ \\ (y {}^{2} - 5y - 22) \: {y(y + 4) + 1(y + 4)} \\ \\ (y {}^{2} - 5y - 22) \: (y + 4) \: (y + 1) \: \:$

$x^{2}$