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

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

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

$$ \begin{aligned} P(x) &= -3x^4+6x^3-2 \\ Q(x) &= 7x^4+2x^3-7x^2 \\ \end{aligned} $$

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

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

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

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

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

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

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