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

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

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

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

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

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

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

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

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

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

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