Subtract polynomials: $ \, \color{blue}{ 3x+8x^2+7 } \, $ , $ \, \color{orangered}{ 12x^2-3-6x }$.

$$ \left( \color{blue}{3x+8x^2+7} \right) - \left( \color{orangered}{12x^2-3-6x} \right) = -4x^2+9x+10 $$

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

$$ \begin{aligned} P(x) &= 8x^2+3x+7 \\ Q(x) &= 12x^2-6x-3 \\ \end{aligned} $$

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

$$ \begin{array}{lrrr} & 8x^2 & 3x & 7 \\ \color{red}{ - } & 12x^2 & -6x & -3 \\ \hline \end{array} $$

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

$$ \begin{array}{lrrr} & 8x^2 & 3x & 7 \\ \color{red}{ - } & -12x^2 & 6x & 3 \\ \hline \end{array} $$

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

$$ \begin{array}{lrrr} & 8x^2 & 3x & 7 \\ \color{red}{ - } & -12x^2 & 6x & 3 \\ \hline & 20x^2 & -3x & 4 \end{array} $$

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