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

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

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

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

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

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

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

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

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

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

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