We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{3x^2} & \color{blue}{4x} & \color{blue}{-10} \\ \hline \color{blue}{5x^2} & & & \\ \hline \color{blue}{-3x} & & & \\ \hline \color{blue}{7} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{3x^2} & \color{blue}{4x} & \color{blue}{-10} \\ \hline \color{blue}{5x^2} & \color{orangered}{15x^4} & \color{orangered}{20x^3} & \color{orangered}{-50x^2} \\ \hline \color{blue}{-3x} & \color{orangered}{-9x^3} & \color{orangered}{-12x^2} & \color{orangered}{30x} \\ \hline \color{blue}{7} & \color{orangered}{21x^2} & \color{orangered}{28x} & \color{orangered}{-70} \\ \hline \end{darray} $$

Combine like terms:

$$ 15x^4 + 20x^3-9x^3-50x^2-12x^2 + 21x^2 + 30x + 28x-70 = \\ 15x^4+11x^3-41x^2+58x-70 $$

Polynomial Operations Calculator