Multiply polynomial $ \, \color{blue}{ x^2+10x-25 } \, $ by $ \, \color{orangered}{ x^2+x-42 }$.

$$ \left( \color{blue}{x^2+10x-25} \right) \cdot \left( \color{orangered}{x^2+x-42} \right) = x^4+11x^3-57x^2-445x+1050 $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{x^2} & \color{blue}{10x} & \color{blue}{-25} \\ \hline \color{blue}{x^2} & & & \\ \hline \color{blue}{x} & & & \\ \hline \color{blue}{-42} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{x^2} & \color{blue}{10x} & \color{blue}{-25} \\ \hline \color{blue}{x^2} & \color{orangered}{x^4} & \color{orangered}{10x^3} & \color{orangered}{-25x^2} \\ \hline \color{blue}{x} & \color{orangered}{x^3} & \color{orangered}{10x^2} & \color{orangered}{-25x} \\ \hline \color{blue}{-42} & \color{orangered}{-42x^2} & \color{orangered}{-420x} & \color{orangered}{1050} \\ \hline \end{darray} $$

Combine like terms:

$$ x^4 + 10x^3 + x^3-25x^2 + 10x^2-42x^2-25x-420x + 1050 = \\ x^4+11x^3-57x^2-445x+1050 $$

Polynomial Operations Calculator