Subtract polynomials: $ \, \color{blue}{ 12-11x-5x^5 } \, $ , $ \, \color{orangered}{ 4x^4+8x-4x^5+2x^3-1 }$.

$$ \left( \color{blue}{12-11x-5x^5} \right) - \left( \color{orangered}{4x^4+8x-4x^5+2x^3-1} \right) = -x^5-4x^4-2x^3-19x+13 $$

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

$$ \begin{aligned} P(x) &= -5x^5-11x+12 \\ Q(x) &= -4x^5+4x^4+2x^3+8x-1 \\ \end{aligned} $$

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

$$ \begin{array}{lrrrrr} & -5x^5 & & & -11x & 12 \\ \color{red}{ - } & -4x^5 & 4x^4 & 2x^3 & 8x & -1 \\ \hline \end{array} $$

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

$$ \begin{array}{lrrrrr} & -5x^5 & & & -11x & 12 \\ \color{red}{ - } & 4x^5 & -4x^4 & -2x^3 & -8x & 1 \\ \hline \end{array} $$

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

$$ \begin{array}{lrrrrr} & -5x^5 & & & -11x & 12 \\ \color{red}{ - } & 4x^5 & -4x^4 & -2x^3 & -8x & 1 \\ \hline & -9x^5 & 4x^4 & 2x^3 & -3x & 11 \end{array} $$

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