Subtract polynomials: $ \, \color{blue}{ x^4+2x^3+8x^2 } \, $ , $ \, \color{orangered}{ 6x+x^4-5x^3 }$.

$$ \left( \color{blue}{x^4+2x^3+8x^2} \right) - \left( \color{orangered}{6x+x^4-5x^3} \right) = 7x^3+8x^2-6x $$

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

$$ \begin{aligned} P(x) &= x^4+2x^3+8x^2 \\ Q(x) &= x^4-5x^3+6x \\ \end{aligned} $$

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

$$ \begin{array}{lrrrr} & x^4 & 2x^3 & 8x^2 & \\ \color{red}{ - } & x^4 & -5x^3 & & 6x \\ \hline \end{array} $$

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

$$ \begin{array}{lrrrr} & x^4 & 2x^3 & 8x^2 & \\ \color{red}{ - } & -x^4 & 5x^3 & & -6x \\ \hline \end{array} $$

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

$$ \begin{array}{lrrrr} & x^4 & 2x^3 & 8x^2 & \\ \color{red}{ - } & -x^4 & 5x^3 & & -6x \\ \hline & 2x^4 & -3x^3 & 8x^2 & 6x \end{array} $$

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