Subtract polynomials: $ \, \color{blue}{ 5x^3+3x^2+5 } \, $ , $ \, \color{orangered}{ 7x^2-9x^2+8x-5 }$.

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

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

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

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

$$ \begin{array}{lrrrr} & 5x^3 & 3x^2 & & 5 \\ \color{red}{ - } & & -2x^2 & 8x & -5 \\ \hline \end{array} $$

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

$$ \begin{array}{lrrrr} & 5x^3 & 3x^2 & & 5 \\ \color{red}{ - } & & 2x^2 & -8x & 5 \\ \hline \end{array} $$

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

$$ \begin{array}{lrrrr} & 5x^3 & 3x^2 & & 5 \\ \color{red}{ - } & & 2x^2 & -8x & 5 \\ \hline & 5x^3 & x^2 & 8x & \end{array} $$

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