Subtract polynomials: $ \, \color{blue}{ 5x+x^4 } \, $ , $ \, \color{orangered}{ 5x^2+x^3 }$.

$$ \left( \color{blue}{5x+x^4} \right) - \left( \color{orangered}{5x^2+x^3} \right) = x^4-x^3-5x^2+5x $$

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

$$ \begin{aligned} P(x) &= x^4+5x \\ Q(x) &= x^3+5x^2 \\ \end{aligned} $$

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

$$ \begin{array}{lrrrr} & x^4 & & & 5x \\ \color{red}{ - } & & x^3 & 5x^2 & \\ \hline \end{array} $$

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

$$ \begin{array}{lrrrr} & x^4 & & & 5x \\ \color{red}{ - } & & -x^3 & -5x^2 & \\ \hline \end{array} $$

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

$$ \begin{array}{lrrrr} & x^4 & & & 5x \\ \color{red}{ - } & & -x^3 & -5x^2 & \\ \hline & x^4 & x^3 & 5x^2 & 5x \end{array} $$

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