Factorize each of the following polynomials:$4x^3 + 20x^2 + 33x + 18$ given that $2x + 3$ is a factor.

Given:

Given expression is $4x^3 + 20x^2 + 33x + 18$ and $2x + 3$ is a factor.

To do:

We have to factorize the given polynomial.

Solution:

Let $f(x)=4x^3 + 20x^2 + 33x + 18$

Dividing $f(x)$ by $2x+3$, we get,

$2x + 3$) $4x ^ { 3 } + 20 x ^ { 2 } + 3 3 x + 18$ ( $2x ^ { 2 } + 7 x +6$

                 $4x^3+6x^2$

        ------------------------------------

                           $14x^2+33x+18$

                           $14x^2+21x$

                       --------------------------

                                      $12x+18$

                                     $12x+18$

                               ------------------

                                            0

$f(x)=(2x+3)(2x^{2}+7 x+6)$

$=(2x+3)(2x^{2}+4 x+3x+6)$

$=(2x+3)[2x(x+2)+3(x+2)]$

$=(2x+3)(2x+3)(x+2)$

Hence, $4x^3 + 20x^2 + 33x + 18=(2x+3)(2x+3)(x+2)$.