Subtract polynomials: $ \, \color{blue}{ 8x^2-3x } \, $ , $ \, \color{orangered}{ 5x-40x^3 }$.

$$ \left( \color{blue}{8x^2-3x} \right) - \left( \color{orangered}{5x-40x^3} \right) = 40x^3+8x^2-8x $$

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

$$ \begin{aligned} P(x) &= 8x^2-3x \\ Q(x) &= -40x^3+5x \\ \end{aligned} $$

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

$$ \begin{array}{lrrr} & & 8x^2 & -3x \\ \color{red}{ - } & -40x^3 & & 5x \\ \hline \end{array} $$

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

$$ \begin{array}{lrrr} & & 8x^2 & -3x \\ \color{red}{ - } & 40x^3 & & -5x \\ \hline \end{array} $$

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

$$ \begin{array}{lrrr} & & 8x^2 & -3x \\ \color{red}{ - } & 40x^3 & & -5x \\ \hline & -40x^3 & 8x^2 & 2x \end{array} $$

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