Subtract polynomials: $ \, \color{blue}{ 7x^5+4x^3-2x^2+12x+5 } \, $ , $ \, \color{orangered}{ 6x^4-9x^3+x^2-3 }$.

$$ \left( \color{blue}{7x^5+4x^3-2x^2+12x+5} \right) - \left( \color{orangered}{6x^4-9x^3+x^2-3} \right) = 7x^5-6x^4+13x^3-3x^2+12x+8 $$

Step 1: Write second polynomial below the first one. Making sure like terms are lined up.

$$ \begin{array}{lrrrrrr} & 7x^5 & & 4x^3 & -2x^2 & 12x & 5 \\ \color{red}{ - } & & 6x^4 & -9x^3 & x^2 & & -3 \\ \hline \end{array} $$

Step 2: Change ALL the signs in the second row.

$$ \begin{array}{lrrrrrr} & 7x^5 & & 4x^3 & -2x^2 & 12x & 5 \\ \color{red}{ - } & & -6x^4 & 9x^3 & -x^2 & & 3 \\ \hline \end{array} $$

Step 3: Combine like terms (add each column):

$$ \begin{array}{lrrrrrr} & 7x^5 & & 4x^3 & -2x^2 & 12x & 5 \\ \color{red}{ - } & & -6x^4 & 9x^3 & -x^2 & & 3 \\ \hline & 7x^5 & 6x^4 & -5x^3 & -x^2 & 12x & 2 \end{array} $$

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