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

$$ \begin{aligned} P(x) &= 2x^9+x^5-8x^4+12x \\ Q(x) &= 6x^{10}+11x^7+8x^5+2x^4 \\ \end{aligned} $$

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

$$ \begin{array}{lrrrrrr} & & 2x^9 & & x^5 & -8x^4 & 12x \\ \color{red}{ - } & 6x^{10} & & 11x^7 & 8x^5 & 2x^4 & \\ \hline \end{array} $$

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

$$ \begin{array}{lrrrrrr} & & 2x^9 & & x^5 & -8x^4 & 12x \\ \color{red}{ - } & -6x^{10} & & -11x^7 & -8x^5 & -2x^4 & \\ \hline \end{array} $$

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

$$ \begin{array}{lrrrrrr} & & 2x^9 & & x^5 & -8x^4 & 12x \\ \color{red}{ - } & -6x^{10} & & -11x^7 & -8x^5 & -2x^4 & \\ \hline & 6x^{10} & 2x^9 & 11x^7 & 9x^5 & -6x^4 & 12x \end{array} $$

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