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

$$ \left( \color{blue}{3x^3+3x^2+9} \right) - \left( \color{orangered}{5x^3-7x^2+6x-9} \right) = -2x^3+10x^2-6x+18 $$

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

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

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

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

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

$$ \begin{array}{lrrrr} & 3x^3 & 3x^2 & & 9 \\ \color{red}{ - } & -5x^3 & 7x^2 & -6x & 9 \\ \hline & 8x^3 & -4x^2 & 6x & \end{array} $$

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