Multiply polynomial $ \, \color{blue}{ x^2+x-6 } \, $ by $ \, \color{orangered}{ x^2-6x+9 }$.

$$ \left( \color{blue}{x^2+x-6} \right) \cdot \left( \color{orangered}{x^2-6x+9} \right) = x^4-5x^3-3x^2+45x-54 $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{x^2} & \color{blue}{x} & \color{blue}{-6} \\ \hline \color{blue}{x^2} & & & \\ \hline \color{blue}{-6x} & & & \\ \hline \color{blue}{9} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{x^2} & \color{blue}{x} & \color{blue}{-6} \\ \hline \color{blue}{x^2} & \color{orangered}{x^4} & \color{orangered}{x^3} & \color{orangered}{-6x^2} \\ \hline \color{blue}{-6x} & \color{orangered}{-6x^3} & \color{orangered}{-6x^2} & \color{orangered}{36x} \\ \hline \color{blue}{9} & \color{orangered}{9x^2} & \color{orangered}{9x} & \color{orangered}{-54} \\ \hline \end{darray} $$

Combine like terms:

$$ x^4 + x^3-6x^3-6x^2-6x^2 + 9x^2 + 36x + 9x-54 = \\ x^4-5x^3-3x^2+45x-54 $$

Polynomial Operations Calculator