# How do you simplify 12a (a^2 + b) - 5b (b^2 + a) + 9b (b^2 + a) - 3a (a^2 + b)?

Sep 20, 2017

$9 {a}^{3} + 13 a b + 4 {b}^{3}$

#### Explanation:

Step one : Expand the brackets

$12 a \left({a}^{2} + b\right) - 5 b \left({b}^{2} + a\right) + 9 b \left({b}^{2} + a\right) - 3 a \left({a}^{2} + b\right) =$

$\left(12 a\right) \left({a}^{2}\right) + \left(12 a\right) \left(b\right) \left(- 5 b\right) \left({b}^{2}\right) + \left(- 5 b\right) \left(a\right) + \left(9 b\right) \left({b}^{2}\right) + \left(9 b\right) \left(a\right) \left(- 3 a\right) \left({a}^{2}\right) + \left(- 3 a\right) \left(b\right) =$
which gives us

$12 {a}^{3} + 12 a b - 5 {b}^{3} - 5 a b + 9 {b}^{3} + 9 a b - 3 {a}^{3} - 3 a b$

Step two : Collect like terms

1. color(red)(12a^3)-5b^3-5ab+9b^3+9ab color(red)( -3a^3)-3ab= color(green)(9a^3)-5b^3-5ab+9b^3+9ab-3ab
2. $9 {a}^{3} + \textcolor{red}{12 a b} - 5 {b}^{3} \textcolor{red}{- 5 a b} + 9 {b}^{3} \textcolor{red}{+ 9 a b} \textcolor{red}{- 3 a b} = 9 {a}^{3} + \textcolor{g r e e n}{13 a b} - 5 {b}^{3} + 9 {b}^{3}$
3. $9 {a}^{3} + 13 a b \textcolor{red}{- 5 {b}^{3}} + \textcolor{red}{9 {b}^{3}} = 9 {a}^{3} + 13 a b + \textcolor{g r e e n}{4 {b}^{3}}$

$9 {a}^{3} + 13 a b + 4 {b}^{3}$