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

$$ \begin{aligned} P(x) &= -n^4+8n^3-8n \\ Q(x) &= 8n^4+5n^3-8n \\ \end{aligned} $$

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

$$ \begin{array}{lrrr} & -n^4 & 8n^3 & -8n \\ \color{red}{ - } & 8n^4 & 5n^3 & -8n \\ \hline \end{array} $$

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

$$ \begin{array}{lrrr} & -n^4 & 8n^3 & -8n \\ \color{red}{ - } & -8n^4 & -5n^3 & 8n \\ \hline \end{array} $$

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

$$ \begin{array}{lrrr} & -n^4 & 8n^3 & -8n \\ \color{red}{ - } & -8n^4 & -5n^3 & 8n \\ \hline & 7n^4 & 13n^3 & -16n \end{array} $$

Polynomial Operations Calculator