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

Jan 14, 2017

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

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

#### 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))
$\textcolor{red}{10 {n}^{4}} \textcolor{w h i t e}{\text{X")color(blue)(+11n^3)color(white)("X")color(green)(-52n^2)color(white)("X")color(magenta)(-12n)color(white)("X}} \textcolor{\mathmr{and} a n \ge}{+ 48}$