Multiply polynomial $ \, \color{blue}{ x^2-3x^3-7x-14 } \, $ by $ \, \color{orangered}{ x-4 }$.

$$ \left( \color{blue}{x^2-3x^3-7x-14} \right) \cdot \left( \color{orangered}{x-4} \right) = -3x^4+13x^3-11x^2+14x+56 $$

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

$$ \begin{aligned} P(x) &= -3x^3+x^2-7x-14 \\ Q(x) &= x-4 \\ \end{aligned} $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|c|}\hline & \color{blue}{-3x^3} & \color{blue}{x^2} & \color{blue}{-7x} & \color{blue}{-14} \\ \hline \color{blue}{x} & & & & \\ \hline \color{blue}{-4} & & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|c|}\hline & \color{blue}{-3x^3} & \color{blue}{x^2} & \color{blue}{-7x} & \color{blue}{-14} \\ \hline \color{blue}{x} & \color{orangered}{-3x^4} & \color{orangered}{x^3} & \color{orangered}{-7x^2} & \color{orangered}{-14x} \\ \hline \color{blue}{-4} & \color{orangered}{12x^3} & \color{orangered}{-4x^2} & \color{orangered}{28x} & \color{orangered}{56} \\ \hline \end{darray} $$

Combine like terms:

$$ -3x^4 + x^3 + 12x^3-7x^2-4x^2-14x + 28x + 56 = \\ -3x^4+13x^3-11x^2+14x+56 $$

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