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

$$ \left( \color{blue}{2u^2+5} \right) \cdot \left( \color{orangered}{5u-2u^2+2} \right) = -4u^4+10u^3-6u^2+25u+10 $$

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

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

We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{-2u^2} & \color{blue}{5u} & \color{blue}{2} \\ \hline \color{blue}{2u^2} & \color{orangered}{-4u^4} & \color{orangered}{10u^3} & \color{orangered}{4u^2} \\ \hline \color{blue}{5} & \color{orangered}{-10u^2} & \color{orangered}{25u} & \color{orangered}{10} \\ \hline \end{darray} $$

Combine like terms:

$$ -4u^4 + 10u^3-10u^2 + 4u^2 + 25u + 10 = \\ -4u^4+10u^3-6u^2+25u+10 $$

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