Multiply polynomial $ \, \color{blue}{ 6x^2+2x+1 } \, $ by $ \, \color{orangered}{ 4x^2+7x-7 }$.

$$ \left( \color{blue}{6x^2+2x+1} \right) \cdot \left( \color{orangered}{4x^2+7x-7} \right) = 24x^4+50x^3-24x^2-7x-7 $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{6x^2} & \color{blue}{2x} & \color{blue}{1} \\ \hline \color{blue}{4x^2} & & & \\ \hline \color{blue}{7x} & & & \\ \hline \color{blue}{-7} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{6x^2} & \color{blue}{2x} & \color{blue}{1} \\ \hline \color{blue}{4x^2} & \color{orangered}{24x^4} & \color{orangered}{8x^3} & \color{orangered}{4x^2} \\ \hline \color{blue}{7x} & \color{orangered}{42x^3} & \color{orangered}{14x^2} & \color{orangered}{7x} \\ \hline \color{blue}{-7} & \color{orangered}{-42x^2} & \color{orangered}{-14x} & \color{orangered}{-7} \\ \hline \end{darray} $$

Combine like terms:

$$ 24x^4 + 8x^3 + 42x^3 + 4x^2 + 14x^2-42x^2 + 7x-14x-7 = \\ 24x^4+50x^3-24x^2-7x-7 $$

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