Multiply polynomial $ \, \color{blue}{ 3x+4 } \, $ by $ \, \color{orangered}{ x-13x^2+x-4 }$.

$$ \left( \color{blue}{3x+4} \right) \cdot \left( \color{orangered}{x-13x^2+x-4} \right) = -39x^3-46x^2-4x-16 $$

Step 1: First we have to write polynomials in descending order.

$$ \begin{aligned} P(x) &= 3x+4 \\ Q(x) &= -13x^2+2x-4 \\ \end{aligned} $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{-13x^2} & \color{blue}{2x} & \color{blue}{-4} \\ \hline \color{blue}{3x} & & & \\ \hline \color{blue}{4} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{-13x^2} & \color{blue}{2x} & \color{blue}{-4} \\ \hline \color{blue}{3x} & \color{orangered}{-39x^3} & \color{orangered}{6x^2} & \color{orangered}{-12x} \\ \hline \color{blue}{4} & \color{orangered}{-52x^2} & \color{orangered}{8x} & \color{orangered}{-16} \\ \hline \end{darray} $$

Combine like terms:

$$ -39x^3 + 6x^2-52x^2-12x + 8x-16 = \\ -39x^3-46x^2-4x-16 $$

Polynomial Operations Calculator