Multiply polynomial $ \, \color{blue}{ 3x^2+4x-2 } \, $ by $ \, \color{orangered}{ x^2-5x+3 }$.

$$ \left( \color{blue}{3x^2+4x-2} \right) \cdot \left( \color{orangered}{x^2-5x+3} \right) = 3x^4-11x^3-13x^2+22x-6 $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{3x^2} & \color{blue}{4x} & \color{blue}{-2} \\ \hline \color{blue}{x^2} & & & \\ \hline \color{blue}{-5x} & & & \\ \hline \color{blue}{3} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{3x^2} & \color{blue}{4x} & \color{blue}{-2} \\ \hline \color{blue}{x^2} & \color{orangered}{3x^4} & \color{orangered}{4x^3} & \color{orangered}{-2x^2} \\ \hline \color{blue}{-5x} & \color{orangered}{-15x^3} & \color{orangered}{-20x^2} & \color{orangered}{10x} \\ \hline \color{blue}{3} & \color{orangered}{9x^2} & \color{orangered}{12x} & \color{orangered}{-6} \\ \hline \end{darray} $$

Combine like terms:

$$ 3x^4 + 4x^3-15x^3-2x^2-20x^2 + 9x^2 + 10x + 12x-6 = \\ 3x^4-11x^3-13x^2+22x-6 $$

Polynomial Operations Calculator