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

$$ \left( \color{blue}{2x^2+3x-6} \right) \cdot \left( \color{orangered}{-5x^2-2x+4} \right) = -10x^4-19x^3+32x^2+24x-24 $$

We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{2x^2} & \color{blue}{3x} & \color{blue}{-6} \\ \hline \color{blue}{-5x^2} & \color{orangered}{-10x^4} & \color{orangered}{-15x^3} & \color{orangered}{30x^2} \\ \hline \color{blue}{-2x} & \color{orangered}{-4x^3} & \color{orangered}{-6x^2} & \color{orangered}{12x} \\ \hline \color{blue}{4} & \color{orangered}{8x^2} & \color{orangered}{12x} & \color{orangered}{-24} \\ \hline \end{darray} $$

Combine like terms:

$$ -10x^4-15x^3-4x^3 + 30x^2-6x^2 + 8x^2 + 12x + 12x-24 = \\ -10x^4-19x^3+32x^2+24x-24 $$

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