Subtract polynomials: $ \, \color{blue}{ 13x^2-6x^5-2x } \, $ , $ \, \color{orangered}{ -10x^2-11x^5+9x }$.

$$ \left( \color{blue}{13x^2-6x^5-2x} \right) - \left( \color{orangered}{-10x^2-11x^5+9x} \right) = 5x^5+23x^2-11x $$

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

$$ \begin{aligned} P(x) &= -6x^5+13x^2-2x \\ Q(x) &= -11x^5-10x^2+9x \\ \end{aligned} $$

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

$$ \begin{array}{lrrr} & -6x^5 & 13x^2 & -2x \\ \color{red}{ - } & -11x^5 & -10x^2 & 9x \\ \hline \end{array} $$

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

$$ \begin{array}{lrrr} & -6x^5 & 13x^2 & -2x \\ \color{red}{ - } & 11x^5 & 10x^2 & -9x \\ \hline \end{array} $$

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

$$ \begin{array}{lrrr} & -6x^5 & 13x^2 & -2x \\ \color{red}{ - } & 11x^5 & 10x^2 & -9x \\ \hline & -17x^5 & 3x^2 & 7x \end{array} $$

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