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