Multiply polynomial $ \, \color{blue}{ 5k^3+3k^2-2k } \, $ by $ \, \color{orangered}{ -8k^3-20k }$.

$$ \left( \color{blue}{5k^3+3k^2-2k} \right) \cdot \left( \color{orangered}{-8k^3-20k} \right) = -40k^6-24k^5-84k^4-60k^3+40k^2 $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{5k^3} & \color{blue}{3k^2} & \color{blue}{-2k} \\ \hline \color{blue}{-8k^3} & & & \\ \hline \color{blue}{-20k} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{5k^3} & \color{blue}{3k^2} & \color{blue}{-2k} \\ \hline \color{blue}{-8k^3} & \color{orangered}{-40k^6} & \color{orangered}{-24k^5} & \color{orangered}{16k^4} \\ \hline \color{blue}{-20k} & \color{orangered}{-100k^4} & \color{orangered}{-60k^3} & \color{orangered}{40k^2} \\ \hline \end{darray} $$

Combine like terms:

$$ -40k^6-24k^5-100k^4 + 16k^4-60k^3 + 40k^2 = \\ -40k^6-24k^5-84k^4-60k^3+40k^2 $$

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