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

$$ \begin{aligned} P(x) &= 2x^3-x+5 \\ Q(x) &= -x^2+3x+1 \\ \end{aligned} $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{2x^3} & \color{blue}{-x} & \color{blue}{5} \\ \hline \color{blue}{-x^2} & & & \\ \hline \color{blue}{3x} & & & \\ \hline \color{blue}{1} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{2x^3} & \color{blue}{-x} & \color{blue}{5} \\ \hline \color{blue}{-x^2} & \color{orangered}{-2x^5} & \color{orangered}{x^3} & \color{orangered}{-5x^2} \\ \hline \color{blue}{3x} & \color{orangered}{6x^4} & \color{orangered}{-3x^2} & \color{orangered}{15x} \\ \hline \color{blue}{1} & \color{orangered}{2x^3} & \color{orangered}{-x} & \color{orangered}{5} \\ \hline \end{darray} $$

Combine like terms:

$$ -2x^5 + x^3 + 6x^4-5x^2-3x^2 + 2x^3 + 15x-x + 5 = \\ -2x^5+6x^4+3x^3-8x^2+14x+5 $$

Polynomial Operations Calculator