# How do you multiply (10x^2-13xy-3y^2)/(8x^2-10xy-3y^2)*(2y+8x)/(2x^2+2y^2)?

Apr 11, 2018

$\frac{40 {x}^{3} - 42 {x}^{2} y - 25 x {y}^{2} - 3 {y}^{3}}{8 {x}^{4} - 10 {x}^{3} y + 5 {x}^{2} {y}^{2} - 10 x {y}^{3} - 3 {y}^{4}}$

#### Explanation:

$\frac{\left(10 {x}^{2} - 13 x y - 3 {y}^{2}\right) \left(2 y + 8 x\right)}{\left(8 {x}^{2} - 10 x y - 3 {y}^{2}\right) \left(2 {x}^{2} + 2 {y}^{2}\right)} =$

$\frac{2 y \left(10 {x}^{2} - 13 x y - 3 {y}^{2}\right) + 8 x \left(10 {x}^{2} - 13 x y - 3 {y}^{2}\right)}{2 {x}^{2} \left(8 {x}^{2} - 10 x y - 3 {y}^{2}\right) + 2 {y}^{2} \left(8 {x}^{2} - 10 x y - 3 {y}^{2}\right)}$

$\frac{2 y 10 {x}^{2} - 2 y 13 x y - 2 y 3 {y}^{2} + 8 x 10 {x}^{2} - 8 x 13 x y - 8 x 3 {y}^{2}}{2 {x}^{2} 8 {x}^{2} - 2 {x}^{2} 10 x y - 2 {x}^{2} 3 {y}^{2} + 2 {y}^{2} 8 {x}^{2} - 2 {y}^{2} 10 x y - 2 {y}^{2} 3 {y}^{2}}$

This can be simplified to:

$\frac{10 {x}^{2} y - 13 x {y}^{2} - 3 {y}^{3} + 40 {x}^{3} - 52 {x}^{2} y - 12 x {y}^{2}}{8 {x}^{4} - 10 {x}^{3} y - 3 {x}^{2} {y}^{2} + 8 {x}^{2} {y}^{2} - 10 x {y}^{3} - 3 {y}^{4}}$

and further:

$\frac{40 {x}^{3} - 42 {x}^{2} y - 25 x {y}^{2} - 3 {y}^{3}}{8 {x}^{4} - 10 {x}^{3} y + 5 {x}^{2} {y}^{2} - 10 x {y}^{3} - 3 {y}^{4}}$

Apr 11, 2018

color(magenta)(80x^3-84x^2y-50xy^2-6y^3)/(color(blue)(16x^4-20x^3y+10x^2y^2-20xy^3-6y^4)

#### Explanation:

$\frac{10 {x}^{2} - 13 x y - 3 {y}^{2}}{8 {x}^{2} - 10 x y - 3 {y}^{2}} \cdot \frac{2 y + 8 x}{2 {x}^{2} + 2 {y}^{2}}$

$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$10 {x}^{2} - 13 x y - 3 {y}^{2}$
$\textcolor{w h i t e}{a a a a a a a a a a a}$$\times \underline{2 y + 8 x}$
$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$20 {x}^{2} y - 26 x {y}^{2} - 6 {y}^{3}$
$\textcolor{w h i t e}{a a a a a a a a a a a}$$- 104 {x}^{2} y - 24 x {y}^{2} + 0 + 80 {x}^{3}$
$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$\overline{- 84 {x}^{2} y - 50 x {y}^{2} - 6 {y}^{3} + 80 {x}^{3}}$

$\textcolor{w h i t e}{a a a a a a a a a a a a a}$color(magenta)(80x^3-84x^2y-50xy^2-6y^3

$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$8 {x}^{2} - 10 x y - 3 {y}^{2}$
$\textcolor{w h i t e}{a a a a a a a a a a a}$$\times \underline{2 {x}^{2} + 2 {y}^{2}}$
$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$16 {x}^{4} - 20 {x}^{3} y - 6 {x}^{2} {y}^{2}$
$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a a a a a a a a a a}$$16 {x}^{2} {y}^{2} - 20 x {y}^{3} - 6 {y}^{4}$
$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$\overline{16 {x}^{4} - 20 {x}^{3} y + 10 {x}^{2} {y}^{2} - 20 x {y}^{3} - 6 {y}^{4}}$

$\textcolor{w h i t e}{a a a a a a a a a a a a a}$color(blue)(16x^4-20x^3y+10x^2y^2-20xy^3-6y^4

$\textcolor{w h i t e}{a a a a a a a a a a a a a}$color(magenta)((80x^3-84x^2y-50xy^2-6y^3)/color(blue)(16x^4-20x^3y+10x^2y^2-20xy^3-6y^4