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

$$ \begin{aligned} P(x) &= x^2+4x-21 \\ Q(x) &= -4x^2-5x+10 \\ \end{aligned} $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{x^2} & \color{blue}{4x} & \color{blue}{-21} \\ \hline \color{blue}{-4x^2} & & & \\ \hline \color{blue}{-5x} & & & \\ \hline \color{blue}{10} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{x^2} & \color{blue}{4x} & \color{blue}{-21} \\ \hline \color{blue}{-4x^2} & \color{orangered}{-4x^4} & \color{orangered}{-16x^3} & \color{orangered}{84x^2} \\ \hline \color{blue}{-5x} & \color{orangered}{-5x^3} & \color{orangered}{-20x^2} & \color{orangered}{105x} \\ \hline \color{blue}{10} & \color{orangered}{10x^2} & \color{orangered}{40x} & \color{orangered}{-210} \\ \hline \end{darray} $$

Combine like terms:

$$ -4x^4-16x^3-5x^3 + 84x^2-20x^2 + 10x^2 + 105x + 40x-210 = \\ -4x^4-21x^3+74x^2+145x-210 $$

Polynomial Operations Calculator