Add $ \, \color{blue}{ x^4-2x^3-7x^3+8x+12 } \, $ and $ \, \color{orangered}{ x^4-2x^3-7x^3+8x+12 }$.

$$ \left( \color{blue}{x^4-2x^3-7x^3+8x+12} \right) + \left( \color{orangered}{x^4-2x^3-7x^3+8x+12} \right) = 2x^4-18x^3+16x+24 $$

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

$$ \begin{aligned} P(x) &= x^4-9x^3+8x+12 \\ Q(x) &= x^4-9x^3+8x+12 \\ \end{aligned} $$

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

$$ \begin{array}{lrrrr} & x^4 & -9x^3 & 8x & 12 \\ \color{blue}{ + } & x^4 & -9x^3 & 8x & 12 \\ \hline \end{array} $$

Step 3: Combine like terms (add each column):

$$ \begin{array}{lrrrr} & x^4 & -9x^3 & 8x & 12 \\ \color{blue}{ + } & x^4 & -9x^3 & 8x & 12 \\ \hline & 2x^4 & -18x^3 & 16x & 24 \end{array} $$

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