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

$$ \begin{aligned} P(x) &= x^2+2x+1 \\ Q(x) &= 4x^2+77 \\ \end{aligned} $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{x^2} & \color{blue}{2x} & \color{blue}{1} \\ \hline \color{blue}{4x^2} & & & \\ \hline \color{blue}{77} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{x^2} & \color{blue}{2x} & \color{blue}{1} \\ \hline \color{blue}{4x^2} & \color{orangered}{4x^4} & \color{orangered}{8x^3} & \color{orangered}{4x^2} \\ \hline \color{blue}{77} & \color{orangered}{77x^2} & \color{orangered}{154x} & \color{orangered}{77} \\ \hline \end{darray} $$

Combine like terms:

$$ 4x^4 + 8x^3 + 77x^2 + 4x^2 + 154x + 77 = \\ 4x^4+8x^3+81x^2+154x+77 $$

Polynomial Operations Calculator