We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|c|}\hline & \color{blue}{4x^4} & \color{blue}{-69x^2} & \color{blue}{-134x} & \color{blue}{-105} \\ \hline \color{blue}{3x} & & & & \\ \hline \color{blue}{-1} & & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|c|}\hline & \color{blue}{4x^4} & \color{blue}{-69x^2} & \color{blue}{-134x} & \color{blue}{-105} \\ \hline \color{blue}{3x} & \color{orangered}{12x^5} & \color{orangered}{-207x^3} & \color{orangered}{-402x^2} & \color{orangered}{-315x} \\ \hline \color{blue}{-1} & \color{orangered}{-4x^4} & \color{orangered}{69x^2} & \color{orangered}{134x} & \color{orangered}{105} \\ \hline \end{darray} $$

Combine like terms:

$$ 12x^5-207x^3-4x^4-402x^2 + 69x^2-315x + 134x + 105 = \\ 12x^5-4x^4-207x^3-333x^2-181x+105 $$

Polynomial Operations Calculator