Subtract polynomials: $ \, \color{blue}{ b+4b^2-3 } \, $ , $ \, \color{orangered}{ -2b^2+2b^3-9 }$.

$$ \left( \color{blue}{b+4b^2-3} \right) - \left( \color{orangered}{-2b^2+2b^3-9} \right) = -2b^3+6b^2+b+6 $$

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

$$ \begin{aligned} P(x) &= 4b^2+b-3 \\ Q(x) &= 2b^3-2b^2-9 \\ \end{aligned} $$

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

$$ \begin{array}{lrrrr} & & 4b^2 & b & -3 \\ \color{red}{ - } & 2b^3 & -2b^2 & & -9 \\ \hline \end{array} $$

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

$$ \begin{array}{lrrrr} & & 4b^2 & b & -3 \\ \color{red}{ - } & -2b^3 & 2b^2 & & 9 \\ \hline \end{array} $$

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

$$ \begin{array}{lrrrr} & & 4b^2 & b & -3 \\ \color{red}{ - } & -2b^3 & 2b^2 & & 9 \\ \hline & 2b^3 & 2b^2 & b & -12 \end{array} $$

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