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

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

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

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

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

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

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

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

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

$$ \begin{array}{lrrr} & -2x^3 & x & \\ \color{red}{ - } & 7x^3 & -7x & 3 \\ \hline & -9x^3 & 8x & -3 \end{array} $$

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