Multiply polynomial $ \, \color{blue}{ x^4+7 } \, $ by $ \, \color{orangered}{ 5x^2+10x-1 }$.

$$ \left( \color{blue}{x^4+7} \right) \cdot \left( \color{orangered}{5x^2+10x-1} \right) = 5x^6+10x^5-x^4+35x^2+70x-7 $$

We can multiply polynomials by using a GRID METHOD

Write one of the polynomials across the top and the other down the left side.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{5x^2} & \color{blue}{10x} & \color{blue}{-1} \\ \hline \color{blue}{x^4} & & & \\ \hline \color{blue}{7} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{5x^2} & \color{blue}{10x} & \color{blue}{-1} \\ \hline \color{blue}{x^4} & \color{orangered}{5x^6} & \color{orangered}{10x^5} & \color{orangered}{-x^4} \\ \hline \color{blue}{7} & \color{orangered}{35x^2} & \color{orangered}{70x} & \color{orangered}{-7} \\ \hline \end{darray} $$

Combine like terms:

$$ 5x^6 + 10x^5 + 35x^2-x^4 + 70x-7 = \\ 5x^6+10x^5-x^4+35x^2+70x-7 $$

Polynomial Operations Calculator