Multiply polynomial $ \, \color{blue}{ 5x^2-6x-3 } \, $ by $ \, \color{orangered}{ 3x^2+7x+2 }$.

$$ \left( \color{blue}{5x^2-6x-3} \right) \cdot \left( \color{orangered}{3x^2+7x+2} \right) = 15x^4+17x^3-41x^2-33x-6 $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{5x^2} & \color{blue}{-6x} & \color{blue}{-3} \\ \hline \color{blue}{3x^2} & & & \\ \hline \color{blue}{7x} & & & \\ \hline \color{blue}{2} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{5x^2} & \color{blue}{-6x} & \color{blue}{-3} \\ \hline \color{blue}{3x^2} & \color{orangered}{15x^4} & \color{orangered}{-18x^3} & \color{orangered}{-9x^2} \\ \hline \color{blue}{7x} & \color{orangered}{35x^3} & \color{orangered}{-42x^2} & \color{orangered}{-21x} \\ \hline \color{blue}{2} & \color{orangered}{10x^2} & \color{orangered}{-12x} & \color{orangered}{-6} \\ \hline \end{darray} $$

Combine like terms:

$$ 15x^4-18x^3 + 35x^3-9x^2-42x^2 + 10x^2-21x-12x-6 = \\ 15x^4+17x^3-41x^2-33x-6 $$

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