Subtract polynomials: $ \, \color{blue}{ 2p^6-4 } \, $ , $ \, \color{orangered}{ 6p^5+8-p^7 }$.

$$ \left( \color{blue}{2p^6-4} \right) - \left( \color{orangered}{6p^5+8-p^7} \right) = p^7+2p^6-6p^5-12 $$

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

$$ \begin{aligned} P(x) &= 2p^6-4 \\ Q(x) &= -p^7+6p^5+8 \\ \end{aligned} $$

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

$$ \begin{array}{lrrrr} & & 2p^6 & & -4 \\ \color{red}{ - } & -p^7 & & 6p^5 & 8 \\ \hline \end{array} $$

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

$$ \begin{array}{lrrrr} & & 2p^6 & & -4 \\ \color{red}{ - } & p^7 & & -6p^5 & -8 \\ \hline \end{array} $$

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

$$ \begin{array}{lrrrr} & & 2p^6 & & -4 \\ \color{red}{ - } & p^7 & & -6p^5 & -8 \\ \hline & -p^7 & 2p^6 & 6p^5 & 4 \end{array} $$

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