Subtract polynomials: $ \, \color{blue}{ 3x^5-2x^4-4x } \, $ , $ \, \color{orangered}{ 4x^2+10x^4-3x }$.

$$ \left( \color{blue}{3x^5-2x^4-4x} \right) - \left( \color{orangered}{4x^2+10x^4-3x} \right) = 3x^5-12x^4-4x^2-x $$

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

$$ \begin{aligned} P(x) &= 3x^5-2x^4-4x \\ Q(x) &= 10x^4+4x^2-3x \\ \end{aligned} $$

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

$$ \begin{array}{lrrrr} & 3x^5 & -2x^4 & & -4x \\ \color{red}{ - } & & 10x^4 & 4x^2 & -3x \\ \hline \end{array} $$

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

$$ \begin{array}{lrrrr} & 3x^5 & -2x^4 & & -4x \\ \color{red}{ - } & & -10x^4 & -4x^2 & 3x \\ \hline \end{array} $$

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

$$ \begin{array}{lrrrr} & 3x^5 & -2x^4 & & -4x \\ \color{red}{ - } & & -10x^4 & -4x^2 & 3x \\ \hline & 3x^5 & 8x^4 & 4x^2 & -7x \end{array} $$

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