Multiply polynomial $ \, \color{blue}{ 2x^3+3x^2 } \, $ by $ \, \color{orangered}{ 4x^4-5x^3-6x^2 }$.

$$ \left( \color{blue}{2x^3+3x^2} \right) \cdot \left( \color{orangered}{4x^4-5x^3-6x^2} \right) = 8x^7+2x^6-27x^5-18x^4 $$

We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{4x^4} & \color{blue}{-5x^3} & \color{blue}{-6x^2} \\ \hline \color{blue}{2x^3} & \color{orangered}{8x^7} & \color{orangered}{-10x^6} & \color{orangered}{-12x^5} \\ \hline \color{blue}{3x^2} & \color{orangered}{12x^6} & \color{orangered}{-15x^5} & \color{orangered}{-18x^4} \\ \hline \end{darray} $$

Combine like terms:

$$ 8x^7-10x^6 + 12x^6-12x^5-15x^5-18x^4 = \\ 8x^7+2x^6-27x^5-18x^4 $$

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