Subtract polynomials: $ \, \color{blue}{ 6x-5x^4 } \, $ , $ \, \color{orangered}{ 4x^4+6x^3+3x }$.

$$ \left( \color{blue}{6x-5x^4} \right) - \left( \color{orangered}{4x^4+6x^3+3x} \right) = -9x^4-6x^3+3x $$

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

$$ \begin{aligned} P(x) &= -5x^4+6x \\ Q(x) &= 4x^4+6x^3+3x \\ \end{aligned} $$

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

$$ \begin{array}{lrrr} & -5x^4 & & 6x \\ \color{red}{ - } & 4x^4 & 6x^3 & 3x \\ \hline \end{array} $$

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

$$ \begin{array}{lrrr} & -5x^4 & & 6x \\ \color{red}{ - } & -4x^4 & -6x^3 & -3x \\ \hline \end{array} $$

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

$$ \begin{array}{lrrr} & -5x^4 & & 6x \\ \color{red}{ - } & -4x^4 & -6x^3 & -3x \\ \hline & -x^4 & 6x^3 & 9x \end{array} $$

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