Subtract polynomials: $ \, \color{blue}{ 10x^4-4x^3-7x^2+5x+9 } \, $ , $ \, \color{orangered}{ 2x^4-5x^3-4x^2+9x+3 }$.

$$ \left( \color{blue}{10x^4-4x^3-7x^2+5x+9} \right) - \left( \color{orangered}{2x^4-5x^3-4x^2+9x+3} \right) = 8x^4+x^3-3x^2-4x+6 $$

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

$$ \begin{array}{lrrrrr} & 10x^4 & -4x^3 & -7x^2 & 5x & 9 \\ \color{red}{ - } & 2x^4 & -5x^3 & -4x^2 & 9x & 3 \\ \hline \end{array} $$

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

$$ \begin{array}{lrrrrr} & 10x^4 & -4x^3 & -7x^2 & 5x & 9 \\ \color{red}{ - } & -2x^4 & 5x^3 & 4x^2 & -9x & -3 \\ \hline \end{array} $$

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

$$ \begin{array}{lrrrrr} & 10x^4 & -4x^3 & -7x^2 & 5x & 9 \\ \color{red}{ - } & -2x^4 & 5x^3 & 4x^2 & -9x & -3 \\ \hline & 12x^4 & -9x^3 & -11x^2 & 14x & 12 \end{array} $$

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