Subtract polynomials: $ \, \color{blue}{ 4x+5x^2-1 } \, $ , $ \, \color{orangered}{ 7x+3+8x^2 }$.

$$ \left( \color{blue}{4x+5x^2-1} \right) - \left( \color{orangered}{7x+3+8x^2} \right) = -3x^2-3x-4 $$

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

$$ \begin{aligned} P(x) &= 5x^2+4x-1 \\ Q(x) &= 8x^2+7x+3 \\ \end{aligned} $$

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

$$ \begin{array}{lrrr} & 5x^2 & 4x & -1 \\ \color{red}{ - } & 8x^2 & 7x & 3 \\ \hline \end{array} $$

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

$$ \begin{array}{lrrr} & 5x^2 & 4x & -1 \\ \color{red}{ - } & -8x^2 & -7x & -3 \\ \hline \end{array} $$

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

$$ \begin{array}{lrrr} & 5x^2 & 4x & -1 \\ \color{red}{ - } & -8x^2 & -7x & -3 \\ \hline & 13x^2 & 11x & 2 \end{array} $$

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