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

$$ \begin{aligned} P(x) &= 11x^3+11x-11 \\ Q(x) &= -8x^4-10x^3+12x-4 \\ \end{aligned} $$

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

$$ \begin{array}{lrrrr} & & 11x^3 & 11x & -11 \\ \color{red}{ - } & -8x^4 & -10x^3 & 12x & -4 \\ \hline \end{array} $$

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

$$ \begin{array}{lrrrr} & & 11x^3 & 11x & -11 \\ \color{red}{ - } & 8x^4 & 10x^3 & -12x & 4 \\ \hline \end{array} $$

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

$$ \begin{array}{lrrrr} & & 11x^3 & 11x & -11 \\ \color{red}{ - } & 8x^4 & 10x^3 & -12x & 4 \\ \hline & -8x^4 & x^3 & 23x & -15 \end{array} $$

Polynomial Operations Calculator