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

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

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

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

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

$$ \begin{array}{lrrr} & 3r^4 & 4r^3 & \\ \color{red}{ - } & r^4 & & -5r \\ \hline \end{array} $$

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

$$ \begin{array}{lrrr} & 3r^4 & 4r^3 & \\ \color{red}{ - } & -r^4 & & 5r \\ \hline \end{array} $$

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

$$ \begin{array}{lrrr} & 3r^4 & 4r^3 & \\ \color{red}{ - } & -r^4 & & 5r \\ \hline & 4r^4 & 4r^3 & -5r \end{array} $$

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