How do you factor 4b^3 +3b^2-16b-12?

May 5, 2015

Regroup to maximize the extractable factors
$4 {b}^{3} + 3 {b}^{2} - 16 - 12 = \textcolor{red}{4 {b}^{3} - 16 b} + \textcolor{b l u e}{3 {b}^{2} - 12}$

$= \textcolor{red}{4 b \left({b}^{2} - 4\right)} + \textcolor{b l u e}{3 \left({b}^{2} - 4\right)}$

$= \left(4 b + 3\right) \left({b}^{2} - 4\right)$

Recognize the second term as the Difference of Squares
$= \left(4 b + 3\right) \left(b + 2\right) \left(b - 2\right)$

May 5, 2015

Collect $4 b$ between the first and third term and $+ 3$ between the second and rourth:
$4 b \left({b}^{2} - 4\right) + 3 \left({b}^{2} - 4\right) =$ collect $\left({b}^{2} - 4\right)$
$\left({b}^{2} - 4\right) \left(4 b + 3\right)$