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

$$ \left( \color{blue}{3x^2+2x-7} \right) \cdot \left( \color{orangered}{5x^2+2x-7} \right) = 15x^4+16x^3-52x^2-28x+49 $$

We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{3x^2} & \color{blue}{2x} & \color{blue}{-7} \\ \hline \color{blue}{5x^2} & \color{orangered}{15x^4} & \color{orangered}{10x^3} & \color{orangered}{-35x^2} \\ \hline \color{blue}{2x} & \color{orangered}{6x^3} & \color{orangered}{4x^2} & \color{orangered}{-14x} \\ \hline \color{blue}{-7} & \color{orangered}{-21x^2} & \color{orangered}{-14x} & \color{orangered}{49} \\ \hline \end{darray} $$

Combine like terms:

$$ 15x^4 + 10x^3 + 6x^3-35x^2 + 4x^2-21x^2-14x-14x + 49 = \\ 15x^4+16x^3-52x^2-28x+49 $$

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