Multiply polynomial $ \, \color{blue}{ 4x^2+x+2 } \, $ by $ \, \color{orangered}{ -3x^2+4x-1 }$.

$$ \left( \color{blue}{4x^2+x+2} \right) \cdot \left( \color{orangered}{-3x^2+4x-1} \right) = -12x^4+13x^3-6x^2+7x-2 $$

We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{4x^2} & \color{blue}{x} & \color{blue}{2} \\ \hline \color{blue}{-3x^2} & \color{orangered}{-12x^4} & \color{orangered}{-3x^3} & \color{orangered}{-6x^2} \\ \hline \color{blue}{4x} & \color{orangered}{16x^3} & \color{orangered}{4x^2} & \color{orangered}{8x} \\ \hline \color{blue}{-1} & \color{orangered}{-4x^2} & \color{orangered}{-x} & \color{orangered}{-2} \\ \hline \end{darray} $$

Combine like terms:

$$ -12x^4-3x^3 + 16x^3-6x^2 + 4x^2-4x^2 + 8x-x-2 = \\ -12x^4+13x^3-6x^2+7x-2 $$

Polynomial Operations Calculator