Multiply polynomial $ \, \color{blue}{ -10x^5+5x^3+6 } \, $ by $ \, \color{orangered}{ 4x^3+2x }$.

$$ \left( \color{blue}{-10x^5+5x^3+6} \right) \cdot \left( \color{orangered}{4x^3+2x} \right) = -40x^8+10x^4+24x^3+12x $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{-10x^5} & \color{blue}{5x^3} & \color{blue}{6} \\ \hline \color{blue}{4x^3} & & & \\ \hline \color{blue}{2x} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{-10x^5} & \color{blue}{5x^3} & \color{blue}{6} \\ \hline \color{blue}{4x^3} & \color{orangered}{-40x^8} & \color{orangered}{20x^6} & \color{orangered}{24x^3} \\ \hline \color{blue}{2x} & \color{orangered}{-20x^6} & \color{orangered}{10x^4} & \color{orangered}{12x} \\ \hline \end{darray} $$

Combine like terms:

$$ -40x^8 + 20x^6-20x^6 + 24x^3 + 10x^4 + 12x = \\ -40x^8+10x^4+24x^3+12x $$

Polynomial Operations Calculator