Multiply polynomial $ \, \color{blue}{ 9x^3-5x^2+7+4 } \, $ by $ \, \color{orangered}{ 9x^2-4x+5 }$.

$$ \left( \color{blue}{9x^3-5x^2+7+4} \right) \cdot \left( \color{orangered}{9x^2-4x+5} \right) = 81x^5-81x^4+65x^3+74x^2-44x+55 $$

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

$$ \begin{aligned} P(x) &= 9x^3-5x^2+11 \\ Q(x) &= 9x^2-4x+5 \\ \end{aligned} $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{9x^3} & \color{blue}{-5x^2} & \color{blue}{11} \\ \hline \color{blue}{9x^2} & & & \\ \hline \color{blue}{-4x} & & & \\ \hline \color{blue}{5} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{9x^3} & \color{blue}{-5x^2} & \color{blue}{11} \\ \hline \color{blue}{9x^2} & \color{orangered}{81x^5} & \color{orangered}{-45x^4} & \color{orangered}{99x^2} \\ \hline \color{blue}{-4x} & \color{orangered}{-36x^4} & \color{orangered}{20x^3} & \color{orangered}{-44x} \\ \hline \color{blue}{5} & \color{orangered}{45x^3} & \color{orangered}{-25x^2} & \color{orangered}{55} \\ \hline \end{darray} $$

Combine like terms:

$$ 81x^5-45x^4-36x^4 + 99x^2 + 20x^3 + 45x^3-44x-25x^2 + 55 = \\ 81x^5-81x^4+65x^3+74x^2-44x+55 $$

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