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

$$ \left( \color{blue}{4x^2+3x+4} \right) \cdot \left( \color{orangered}{3x^2-4x-3} \right) = 12x^4-7x^3-12x^2-25x-12 $$

We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{4x^2} & \color{blue}{3x} & \color{blue}{4} \\ \hline \color{blue}{3x^2} & \color{orangered}{12x^4} & \color{orangered}{9x^3} & \color{orangered}{12x^2} \\ \hline \color{blue}{-4x} & \color{orangered}{-16x^3} & \color{orangered}{-12x^2} & \color{orangered}{-16x} \\ \hline \color{blue}{-3} & \color{orangered}{-12x^2} & \color{orangered}{-9x} & \color{orangered}{-12} \\ \hline \end{darray} $$

Combine like terms:

$$ 12x^4 + 9x^3-16x^3 + 12x^2-12x^2-12x^2-16x-9x-12 = \\ 12x^4-7x^3-12x^2-25x-12 $$

Polynomial Operations Calculator