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

$$ \begin{aligned} P(x) &= -8x^3+3x^2-7 \\ Q(x) &= x^3+4x^2-8 \\ \end{aligned} $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{-8x^3} & \color{blue}{3x^2} & \color{blue}{-7} \\ \hline \color{blue}{x^3} & & & \\ \hline \color{blue}{4x^2} & & & \\ \hline \color{blue}{-8} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{-8x^3} & \color{blue}{3x^2} & \color{blue}{-7} \\ \hline \color{blue}{x^3} & \color{orangered}{-8x^6} & \color{orangered}{3x^5} & \color{orangered}{-7x^3} \\ \hline \color{blue}{4x^2} & \color{orangered}{-32x^5} & \color{orangered}{12x^4} & \color{orangered}{-28x^2} \\ \hline \color{blue}{-8} & \color{orangered}{64x^3} & \color{orangered}{-24x^2} & \color{orangered}{56} \\ \hline \end{darray} $$

Combine like terms:

$$ -8x^6 + 3x^5-32x^5-7x^3 + 12x^4 + 64x^3-28x^2-24x^2 + 56 = \\ -8x^6-29x^5+12x^4+57x^3-52x^2+56 $$

Polynomial Operations Calculator