Multiply polynomial $ \, \color{blue}{ 3x^2+2x+1 } \, $ by $ \, \color{orangered}{ 3x^2+2x+1 }$.

$$ \left( \color{blue}{3x^2+2x+1} \right) \cdot \left( \color{orangered}{3x^2+2x+1} \right) = 9x^4+12x^3+10x^2+4x+1 $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{3x^2} & \color{blue}{2x} & \color{blue}{1} \\ \hline \color{blue}{3x^2} & & & \\ \hline \color{blue}{2x} & & & \\ \hline \color{blue}{1} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{3x^2} & \color{blue}{2x} & \color{blue}{1} \\ \hline \color{blue}{3x^2} & \color{orangered}{9x^4} & \color{orangered}{6x^3} & \color{orangered}{3x^2} \\ \hline \color{blue}{2x} & \color{orangered}{6x^3} & \color{orangered}{4x^2} & \color{orangered}{2x} \\ \hline \color{blue}{1} & \color{orangered}{3x^2} & \color{orangered}{2x} & \color{orangered}{1} \\ \hline \end{darray} $$

Combine like terms:

$$ 9x^4 + 6x^3 + 6x^3 + 3x^2 + 4x^2 + 3x^2 + 2x + 2x + 1 = \\ 9x^4+12x^3+10x^2+4x+1 $$

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