Multiply polynomial $ \, \color{blue}{ x^2+2x+1 } \, $ by $ \, \color{orangered}{ x^2+7x-42-6x }$.

$$ \left( \color{blue}{x^2+2x+1} \right) \cdot \left( \color{orangered}{x^2+7x-42-6x} \right) = x^4+3x^3-39x^2-83x-42 $$

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

$$ \begin{aligned} P(x) &= x^2+2x+1 \\ Q(x) &= x^2+x-42 \\ \end{aligned} $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{x^2} & \color{blue}{2x} & \color{blue}{1} \\ \hline \color{blue}{x^2} & & & \\ \hline \color{blue}{x} & & & \\ \hline \color{blue}{-42} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{x^2} & \color{blue}{2x} & \color{blue}{1} \\ \hline \color{blue}{x^2} & \color{orangered}{x^4} & \color{orangered}{2x^3} & \color{orangered}{x^2} \\ \hline \color{blue}{x} & \color{orangered}{x^3} & \color{orangered}{2x^2} & \color{orangered}{x} \\ \hline \color{blue}{-42} & \color{orangered}{-42x^2} & \color{orangered}{-84x} & \color{orangered}{-42} \\ \hline \end{darray} $$

Combine like terms:

$$ x^4 + 2x^3 + x^3 + x^2 + 2x^2-42x^2 + x-84x-42 = \\ x^4+3x^3-39x^2-83x-42 $$

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