Multiply polynomial $ \, \color{blue}{ 9x^3+4x^2+6x+3 } \, $ by $ \, \color{orangered}{ -x^3-x }$.

$$ \left( \color{blue}{9x^3+4x^2+6x+3} \right) \cdot \left( \color{orangered}{-x^3-x} \right) = -9x^6-4x^5-15x^4-7x^3-6x^2-3x $$

We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

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

Combine like terms:

$$ -9x^6-4x^5-9x^4-6x^4-4x^3-3x^3-6x^2-3x = \\ -9x^6-4x^5-15x^4-7x^3-6x^2-3x $$

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