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

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

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

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

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

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

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

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

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

$$ \begin{array}{lrr} & x^4 & 5x \\ \color{red}{ - } & -3x^4 & -4x \\ \hline & 4x^4 & 9x \end{array} $$

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