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

$$ \begin{aligned} P(x) &= 4x^3-5x^2+3x \\ Q(x) &= -x^2+4x \\ \end{aligned} $$

We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{4x^3} & \color{blue}{-5x^2} & \color{blue}{3x} \\ \hline \color{blue}{-x^2} & \color{orangered}{-4x^5} & \color{orangered}{5x^4} & \color{orangered}{-3x^3} \\ \hline \color{blue}{4x} & \color{orangered}{16x^4} & \color{orangered}{-20x^3} & \color{orangered}{12x^2} \\ \hline \end{darray} $$

Combine like terms:

$$ -4x^5 + 5x^4 + 16x^4-3x^3-20x^3 + 12x^2 = \\ -4x^5+21x^4-23x^3+12x^2 $$

Polynomial Operations Calculator