Subtract polynomials: $ \, \color{blue}{ x^6+7x^5+2x^4+9x^3+10x^2+12x+15 } \, $ , $ \, \color{orangered}{ x^6+9x^5+8x^4+9x^3+12x^2+15x+20 }$.

$$ \left( \color{blue}{x^6+7x^5+2x^4+9x^3+10x^2+12x+15} \right) - \left( \color{orangered}{x^6+9x^5+8x^4+9x^3+12x^2+15x+20} \right) = -2x^5-6x^4-2x^2-3x-5 $$

Step 1: Write second polynomial below the first one. Making sure like terms are lined up.

$$ \begin{array}{lrrrrrrr} & x^6 & 7x^5 & 2x^4 & 9x^3 & 10x^2 & 12x & 15 \\ \color{red}{ - } & x^6 & 9x^5 & 8x^4 & 9x^3 & 12x^2 & 15x & 20 \\ \hline \end{array} $$

Step 2: Change ALL the signs in the second row.

$$ \begin{array}{lrrrrrrr} & x^6 & 7x^5 & 2x^4 & 9x^3 & 10x^2 & 12x & 15 \\ \color{red}{ - } & -x^6 & -9x^5 & -8x^4 & -9x^3 & -12x^2 & -15x & -20 \\ \hline \end{array} $$

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

$$ \begin{array}{lrrrrrrr} & x^6 & 7x^5 & 2x^4 & 9x^3 & 10x^2 & 12x & 15 \\ \color{red}{ - } & -x^6 & -9x^5 & -8x^4 & -9x^3 & -12x^2 & -15x & -20 \\ \hline & 2x^6 & 16x^5 & 10x^4 & 18x^3 & 22x^2 & 27x & 35 \end{array} $$

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