# How do you multiply (b^2-4b+6)^2?

Apr 2, 2017

See the entire solution process below:

#### Explanation:

First, we can rewrite this expression as:

$\left(\textcolor{red}{{b}^{2}} - \textcolor{red}{4 b} + \textcolor{red}{6}\right) \left(\textcolor{b l u e}{{b}^{2}} - \textcolor{b l u e}{4 b} + \textcolor{b l u e}{6}\right)$

To multiply these two terms you multiply each individual term in the left parenthesis by each individual term in the right parenthesis.

$\left(\textcolor{red}{{b}^{2}} \times \textcolor{b l u e}{{b}^{2}}\right) - \left(\textcolor{red}{{b}^{2}} \times \textcolor{b l u e}{4 b}\right) + \left(\textcolor{red}{{b}^{2}} \times \textcolor{b l u e}{6}\right) - \left(\textcolor{red}{4 b} \times \textcolor{b l u e}{{b}^{2}}\right) + \left(\textcolor{red}{4 b} \times \textcolor{b l u e}{4 b}\right) - \left(\textcolor{red}{4 b} \times \textcolor{b l u e}{6}\right) + \left(\textcolor{red}{6} \times \textcolor{b l u e}{{b}^{2}}\right) - \left(\textcolor{red}{6} \times \textcolor{b l u e}{4 b}\right) + \left(\textcolor{red}{6} \times \textcolor{b l u e}{6}\right)$

${b}^{4} - 4 {b}^{3} + 6 {b}^{2} - 4 {b}^{3} + 16 {b}^{2} - 24 b + 6 {b}^{2} - 24 b + 36$

We can now group and combine like terms:

${b}^{4} - 4 {b}^{3} - 4 {b}^{3} + 6 {b}^{2} + 16 {b}^{2} + 6 {b}^{2} - 24 b - 24 b + 36$

${b}^{4} + \left(- 4 - 4\right) {b}^{3} + \left(6 + 16 + 6\right) {b}^{2} + \left(- 24 - 24\right) b + 36$

${b}^{4} - 8 {b}^{3} + 28 {b}^{2} - 48 b + 36$

Apr 2, 2017

color(blue)(b^4-8b^3+28b^2-48b+36

#### Explanation:

$\textcolor{w h i t e}{a a a a a a a a a a a a a}$${b}^{2} - 4 b + 6$
$\textcolor{w h i t e}{a a a a a a a a a a a a a}$${b}^{2} - 4 b + 6$
$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$- - - - -$
$\textcolor{w h i t e}{a a a a a a a a a a a a a}$${b}^{4} - 4 {b}^{3} + 6 {b}^{2}$
$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a}$$- 4 {b}^{3} + 16 {b}^{2} - 24 b$
$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a a a a a a a a}$$6 {b}^{2} - 24 b + 36$
$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$- - - - - - - - - -$
$\textcolor{w h i t e}{a a a a a a a a a a a a}$color(blue)(b^4-8b^3+28b^2-48b+36