Subtract polynomials: $ \, \color{blue}{ -18x^2+x-32 } \, $ , $ \, \color{orangered}{ 40-13x^2 }$.

$$ \left( \color{blue}{-18x^2+x-32} \right) - \left( \color{orangered}{40-13x^2} \right) = -5x^2+x-72 $$

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

$$ \begin{aligned} P(x) &= -18x^2+x-32 \\ Q(x) &= -13x^2+40 \\ \end{aligned} $$

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

$$ \begin{array}{lrrr} & -18x^2 & x & -32 \\ \color{red}{ - } & -13x^2 & & 40 \\ \hline \end{array} $$

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

$$ \begin{array}{lrrr} & -18x^2 & x & -32 \\ \color{red}{ - } & 13x^2 & & -40 \\ \hline \end{array} $$

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

$$ \begin{array}{lrrr} & -18x^2 & x & -32 \\ \color{red}{ - } & 13x^2 & & -40 \\ \hline & -31x^2 & x & 8 \end{array} $$

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