Multiply polynomial $ \, \color{blue}{ 10x^2-4+3 } \, $ by $ \, \color{orangered}{ 7x^2-3x+13 }$.

$$ \left( \color{blue}{10x^2-4+3} \right) \cdot \left( \color{orangered}{7x^2-3x+13} \right) = 70x^4-30x^3+123x^2+3x-13 $$

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

$$ \begin{aligned} P(x) &= 10x^2-1 \\ Q(x) &= 7x^2-3x+13 \\ \end{aligned} $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{7x^2} & \color{blue}{-3x} & \color{blue}{13} \\ \hline \color{blue}{10x^2} & & & \\ \hline \color{blue}{-1} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{7x^2} & \color{blue}{-3x} & \color{blue}{13} \\ \hline \color{blue}{10x^2} & \color{orangered}{70x^4} & \color{orangered}{-30x^3} & \color{orangered}{130x^2} \\ \hline \color{blue}{-1} & \color{orangered}{-7x^2} & \color{orangered}{3x} & \color{orangered}{-13} \\ \hline \end{darray} $$

Combine like terms:

$$ 70x^4-30x^3-7x^2 + 130x^2 + 3x-13 = \\ 70x^4-30x^3+123x^2+3x-13 $$

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