# How do you find the product (2n^2+3n-6)(5n^2-2n-8)?

Jan 14, 2017

#### Answer:

Multiply each of the terms inside the left parenthesis by each of the terms inside the right parenthesis, group and then combine like terms. See full process below:

#### Explanation:

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

$\left(\textcolor{red}{2 {n}^{2}} + \textcolor{red}{3 n} - \textcolor{red}{6}\right) \left(\textcolor{b l u e}{5 {n}^{2}} - \textcolor{b l u e}{2 n} - \textcolor{b l u e}{8}\right)$ becomes:

$\left(\textcolor{red}{2 {n}^{2}} \times \textcolor{b l u e}{5 {n}^{2}}\right) - \left(\textcolor{red}{2 {n}^{2}} \times \textcolor{b l u e}{2 n}\right) - \left(\textcolor{red}{2 {n}^{2}} \times \textcolor{b l u e}{8}\right) + \left(\textcolor{red}{3 n} \times \textcolor{b l u e}{5 {n}^{2}}\right) - \left(\textcolor{red}{3 n} \times \textcolor{b l u e}{2 n}\right) - \left(\textcolor{red}{3 n} \times \textcolor{b l u e}{8}\right) - \left(\textcolor{red}{6} \times \textcolor{b l u e}{5 {n}^{2}}\right) + \left(\textcolor{red}{6} \times \textcolor{b l u e}{2 n}\right) + \left(\textcolor{red}{6} \times \textcolor{b l u e}{8}\right)$

$10 {n}^{4} - 4 {n}^{3} - 16 {n}^{2} + 15 {n}^{3} - 6 {n}^{2} - 24 n - 30 {n}^{2} + 12 n + 48$

Next, We can group like terms:

$10 {n}^{4} - 4 {n}^{3} + 15 {n}^{3} - 16 {n}^{2} - 6 {n}^{2} - 30 {n}^{2} - 24 n + 12 n + 48$

Now, we can combine like terms:

$10 {n}^{4} + \left(- 4 + 15\right) {n}^{3} + \left(- 16 - 6 - 30\right) {n}^{2} + \left(- 24 + 12\right) n + 48$

$10 {n}^{4} + 11 {n}^{3} - 52 {n}^{2} - 12 n + 48$

Jan 14, 2017

#### Answer:

$10 {n}^{4} + 11 {n}^{3} - 52 {n}^{2} - 12 n + 48$

#### Explanation:

Using tabular multiplication:

color(white)("XX")underline(color(white)("X")xxcolor(white)("X")|color(white)("XX")2n^2color(white)("X")+3ncolor(white)("XX)-6)
$\textcolor{w h i t e}{\text{XXX")5n^2color(white)("X")|color(white)("X")color(red)(10n^2)color(white)("X")color(blue)(+15n^3)color(white)("X}} \textcolor{g r e e n}{- 30 {n}^{2}}$
$\textcolor{w h i t e}{\text{XX")-2ncolor(white)("X")|color(white)("X")color(blue)(-4n^3)color(white)("X")color(green)(-6n^2)color(white)("X}} \textcolor{m a \ge n t a}{+ 12 n}$
color(white)("Xx")underline(color(white)("X")-8color(white)("X")|color(white)("X")color(green)(-16n^2)color(white)("X")color(magenta)(-24n)color(white)("X")color(orange)(+48))
$\textcolor{red}{10 {n}^{4}} \textcolor{w h i t e}{\text{X")color(blue)(+11n^3)color(white)("X")color(green)(-52n^2)color(white)("X")color(magenta)(-12n)color(white)("X}} \textcolor{\mathmr{and} a n \ge}{+ 48}$