Multiply polynomial $ \, \color{blue}{ 7x+3 } \, $ by $ \, \color{orangered}{ -x^2-5x+8 }$.

$$ \left( \color{blue}{7x+3} \right) \cdot \left( \color{orangered}{-x^2-5x+8} \right) = -7x^3-38x^2+41x+24 $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{-x^2} & \color{blue}{-5x} & \color{blue}{8} \\ \hline \color{blue}{7x} & & & \\ \hline \color{blue}{3} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{-x^2} & \color{blue}{-5x} & \color{blue}{8} \\ \hline \color{blue}{7x} & \color{orangered}{-7x^3} & \color{orangered}{-35x^2} & \color{orangered}{56x} \\ \hline \color{blue}{3} & \color{orangered}{-3x^2} & \color{orangered}{-15x} & \color{orangered}{24} \\ \hline \end{darray} $$

Combine like terms:

$$ -7x^3-35x^2-3x^2 + 56x-15x + 24 = \\ -7x^3-38x^2+41x+24 $$

Polynomial Operations Calculator