Multiply polynomial $ \, \color{blue}{ 3x-4 } \, $ by $ \, \color{orangered}{ 5+3x-4x^2 }$.

$$ \left( \color{blue}{3x-4} \right) \cdot \left( \color{orangered}{5+3x-4x^2} \right) = -12x^3+25x^2+3x-20 $$

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

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

We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{-4x^2} & \color{blue}{3x} & \color{blue}{5} \\ \hline \color{blue}{3x} & \color{orangered}{-12x^3} & \color{orangered}{9x^2} & \color{orangered}{15x} \\ \hline \color{blue}{-4} & \color{orangered}{16x^2} & \color{orangered}{-12x} & \color{orangered}{-20} \\ \hline \end{darray} $$

Combine like terms:

$$ -12x^3 + 9x^2 + 16x^2 + 15x-12x-20 = \\ -12x^3+25x^2+3x-20 $$

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