Question 2: Factorize each of the following expresssions:
\((iii)\) \(\;\) \((2x^2+7x+3)(2x^2+7x+5)+1 \)
Solution:
$\displaystyle\ \ \ \ \left( 2x^{2} +7x+3\right)\left( 2x^{2} +7x+5\right) +1$
Let \(\; y=2x^2+7x\), we have
$\displaystyle =( y+3)( y+5) +1$
$\displaystyle =y( y+5) +3( y+5) +1$
$\displaystyle =y^{2} +5y+3y+15+1$
$\displaystyle =y^{2} +8y+16$
$\displaystyle =y^{2} +2( y)( 4) +4^{2}$
$\displaystyle =( y+4)^{2}$
$\displaystyle =\left( 2x^{2} +7x+4\right)^{2}$