Multiply polynomial $ \, \color{blue}{ 6x^3+x^2+9x+15 } \, $ by $ \, \color{orangered}{ -x^3-x }$.

$$ \left( \color{blue}{6x^3+x^2+9x+15} \right) \cdot \left( \color{orangered}{-x^3-x} \right) = -6x^6-x^5-15x^4-16x^3-9x^2-15x $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|c|}\hline & \color{blue}{6x^3} & \color{blue}{x^2} & \color{blue}{9x} & \color{blue}{15} \\ \hline \color{blue}{-x^3} & & & & \\ \hline \color{blue}{-x} & & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|c|}\hline & \color{blue}{6x^3} & \color{blue}{x^2} & \color{blue}{9x} & \color{blue}{15} \\ \hline \color{blue}{-x^3} & \color{orangered}{-6x^6} & \color{orangered}{-x^5} & \color{orangered}{-9x^4} & \color{orangered}{-15x^3} \\ \hline \color{blue}{-x} & \color{orangered}{-6x^4} & \color{orangered}{-x^3} & \color{orangered}{-9x^2} & \color{orangered}{-15x} \\ \hline \end{darray} $$

Combine like terms:

$$ -6x^6-x^5-6x^4-9x^4-x^3-15x^3-9x^2-15x = \\ -6x^6-x^5-15x^4-16x^3-9x^2-15x $$

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