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

$$ \left( \color{blue}{-3-4x^2-8x^4} \right) - \left( \color{orangered}{-1+3x^4+8x^2} \right) = -11x^4-12x^2-2 $$

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

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

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

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

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

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

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

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

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