Question 2: Factorize each of the following expresssions:
\((iv)\) \(\; \)\((3x^2+5x+3)(3x^2+5x+5)-3 \)
Solution:
$\displaystyle\ \ \ \ \left( 3x^{2} +5x+3\right)\left( 3x^{2} +5x+5\right) -3$
Let \(\; y=3x^2+5x\), we have
$\displaystyle =( y+3)( y+5) -3$
$\displaystyle =y( y+5) +3( y+5) -3$
$\displaystyle =y^{2} +5y+3y+15-3$
$\displaystyle =y^{2} +8y+12$
$\displaystyle =y^{2} +2y+6y+12$
$\displaystyle =y( y+2) +6( y+2)$
$\displaystyle =( y+2)( y+6)$
$\displaystyle =\left( 3x^{2} +5x+2\right)\left( 3x^{2} +5x+6\right)$