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

$$ \begin{aligned} P(x) &= 5x^4-2x^3+7x^2 \\ Q(x) &= 7x^4+7x^3+1 \\ \end{aligned} $$

Step 2: Write second polynomial below the first one. Making sure like terms are lined up.

$$ \begin{array}{lrrrr} & 5x^4 & -2x^3 & 7x^2 & \\ \color{red}{ - } & 7x^4 & 7x^3 & & 1 \\ \hline \end{array} $$

Step 3: Change ALL the signs in the second row.

$$ \begin{array}{lrrrr} & 5x^4 & -2x^3 & 7x^2 & \\ \color{red}{ - } & -7x^4 & -7x^3 & & -1 \\ \hline \end{array} $$

Step 4: Combine like terms (add each column):

$$ \begin{array}{lrrrr} & 5x^4 & -2x^3 & 7x^2 & \\ \color{red}{ - } & -7x^4 & -7x^3 & & -1 \\ \hline & 12x^4 & 5x^3 & 7x^2 & 1 \end{array} $$

Polynomial Operations Calculator