Multiply polynomial $ \, \color{blue}{ 4x^3-7x^2-16x+3 } \, $ by $ \, \color{orangered}{ x-3 }$.

$$ \left( \color{blue}{4x^3-7x^2-16x+3} \right) \cdot \left( \color{orangered}{x-3} \right) = 4x^4-19x^3+5x^2+51x-9 $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|c|}\hline & \color{blue}{4x^3} & \color{blue}{-7x^2} & \color{blue}{-16x} & \color{blue}{3} \\ \hline \color{blue}{x} & & & & \\ \hline \color{blue}{-3} & & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|c|}\hline & \color{blue}{4x^3} & \color{blue}{-7x^2} & \color{blue}{-16x} & \color{blue}{3} \\ \hline \color{blue}{x} & \color{orangered}{4x^4} & \color{orangered}{-7x^3} & \color{orangered}{-16x^2} & \color{orangered}{3x} \\ \hline \color{blue}{-3} & \color{orangered}{-12x^3} & \color{orangered}{21x^2} & \color{orangered}{48x} & \color{orangered}{-9} \\ \hline \end{darray} $$

Combine like terms:

$$ 4x^4-7x^3-12x^3-16x^2 + 21x^2 + 3x + 48x-9 = \\ 4x^4-19x^3+5x^2+51x-9 $$

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