Question
Factorize the following polynomials: $(y+2)(y-3)(y+8)(y+3)+56$
✓
Answer
$(y+2)(y-3)(y+8)(y+3)+56$
$=(y+2)(y+3)(y-3)(y+8)+56$
$=\left(y^2+3 y+2 y+6\right)\left(y^2+8 y-3 y-24\right)+56$
$=\left(y^2+5 y+6\right)\left(y^2+5 y-24\right)+56$
$=(m+6)(m-24)+56 \ldots[\text { Putting } y 2+5 y=m]$
$=m(m-24)+6(m-24)+56$
$=m^2-24 m+6 m-144+56$
$=m^2-18 m-88$
$=m^2-22 m+4 m-88$
$=m(m-22)+4(m-22)$
$=(m-22)(m+4)$
$=\left(y^2+5 y-22\right)\left(y^2+5 y+4\right) \ldots\left[\text { Replacing } m=y^2+5 y\right]$
$=\left(y^2+5 y-22\right)\left(y^2+4 y+y+4\right)$
$=\left(y^2+5 y-22\right)[y(y+4)+1(y+4)]$
$=\left(y^2+5 y-22\right)(y+4)(y+1)$
$=(y+2)(y+3)(y-3)(y+8)+56$
$=\left(y^2+3 y+2 y+6\right)\left(y^2+8 y-3 y-24\right)+56$
$=\left(y^2+5 y+6\right)\left(y^2+5 y-24\right)+56$
$=(m+6)(m-24)+56 \ldots[\text { Putting } y 2+5 y=m]$
$=m(m-24)+6(m-24)+56$
$=m^2-24 m+6 m-144+56$
$=m^2-18 m-88$
$=m^2-22 m+4 m-88$
$=m(m-22)+4(m-22)$
$=(m-22)(m+4)$
$=\left(y^2+5 y-22\right)\left(y^2+5 y+4\right) \ldots\left[\text { Replacing } m=y^2+5 y\right]$
$=\left(y^2+5 y-22\right)\left(y^2+4 y+y+4\right)$
$=\left(y^2+5 y-22\right)[y(y+4)+1(y+4)]$
$=\left(y^2+5 y-22\right)(y+4)(y+1)$
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