# How do you find the product (x^2+5x-1)(5x^2-6x+1)?

Feb 12, 2017

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

$\left(\textcolor{red}{{x}^{2}} + \textcolor{red}{5 x} - \textcolor{red}{1}\right) \left(\textcolor{b l u e}{5 {x}^{2}} - \textcolor{b l u e}{6 x} + \textcolor{b l u e}{1}\right)$ becomes:

$\left(\textcolor{red}{{x}^{2}} \times \textcolor{b l u e}{5 {x}^{2}}\right) - \left(\textcolor{red}{{x}^{2}} \times \textcolor{b l u e}{6 x}\right) + \left(\textcolor{red}{{x}^{2}} \times \textcolor{b l u e}{1}\right) + \left(\textcolor{red}{5 x} \times \textcolor{b l u e}{5 {x}^{2}}\right) - \left(\textcolor{red}{5 x} \times \textcolor{b l u e}{6 x}\right) + \left(\textcolor{red}{5 x} \times \textcolor{b l u e}{1}\right) - \left(\textcolor{red}{1} \times \textcolor{b l u e}{5 {x}^{2}}\right) + \left(\textcolor{red}{1} \times \textcolor{b l u e}{6 x}\right) - \left(\textcolor{red}{1} \times \textcolor{b l u e}{1}\right)$

$5 {x}^{4} - 6 {x}^{3} + {x}^{2} + 25 {x}^{3} - 30 {x}^{2} + 5 x - 5 {x}^{2} + 6 x - 1$

We can now group and combine like terms:

$5 {x}^{4} - 6 {x}^{3} + 25 {x}^{3} + {x}^{2} - 30 {x}^{2} - 5 {x}^{2} + 5 x + 6 x - 1$

$5 {x}^{4} + \left(- 6 + 25\right) {x}^{3} + \left(1 - 30 - 5\right) {x}^{2} + \left(5 + 6\right) x - 1$

$5 {x}^{4} + 19 {x}^{3} - 34 {x}^{2} + 11 x - 1$