# How do you multiply  (2x^2 + y^2)(x - 2y) ?

Jul 5, 2015

By applying the distributive property several times you can get:
$\left(2 {x}^{2} + {y}^{2}\right) \left(x - 2 y\right)$
$\textcolor{w h i t e}{\text{XXXX}}$$= 2 {x}^{3} + x {y}^{2} - 4 {x}^{2} y - 2 {y}^{3}$

#### Explanation:

The distributive property tells us that
$\textcolor{w h i t e}{\text{XXXX}}$$\left(\textcolor{red}{a} + \textcolor{b l u e}{b}\right) \left(\textcolor{g r e e n}{p} - \textcolor{\mathmr{and} a n \ge}{q}\right) = \left(\textcolor{red}{a} + \textcolor{b l u e}{b}\right) \textcolor{g r e e n}{p} - \left(\textcolor{red}{a} + \textcolor{b l u e}{b}\right) \textcolor{\mathmr{and} a n \ge}{q}$
and that
$\textcolor{w h i t e}{\text{XXXX}}$$\left(\textcolor{red}{a} + \textcolor{b l u e}{b}\right) \textcolor{g r e e n}{p} = \textcolor{red}{a} \textcolor{g r e e n}{p} + \textcolor{b l u e}{b} \textcolor{g r e e n}{p}$

Therefore
$\textcolor{w h i t e}{\text{XXXX}}$$\left(\textcolor{red}{2 {x}^{2}} + \textcolor{b l u e}{{y}^{2}}\right) \left(\textcolor{g r e e n}{x} - \textcolor{\mathmr{and} a n \ge}{2 y}\right)$

$\textcolor{w h i t e}{\text{XXXX}}$$\left(\textcolor{red}{2 {x}^{2}} + \textcolor{b l u e}{{y}^{2}}\right) \textcolor{g r e e n}{x} - \left(\textcolor{red}{2 {x}^{2}} + \textcolor{b l u e}{{y}^{2}}\right) \textcolor{\mathmr{and} a n \ge}{2 y}$

$\textcolor{w h i t e}{\text{XXXX}}$$= \left(\textcolor{red}{2 {x}^{2}}\right) \left(\textcolor{g r e e n}{x}\right) + \left(\textcolor{b l u e}{{y}^{2}}\right) \left(\textcolor{g r e e n}{x}\right) - \left(\textcolor{red}{2 {x}^{2}}\right) \left(\textcolor{\mathmr{and} a n \ge}{2 y}\right) - \left(\textcolor{b l u e}{{y}^{2}}\right) \left(\textcolor{\mathmr{and} a n \ge}{2 y}\right)$

$\textcolor{w h i t e}{\text{XXXX}}$$= 2 {x}^{3} + x {y}^{2} - 4 {x}^{2} y - 2 {y}^{3}$