Divide using synthetic division $$ ( 3x^{3}+10x^{2}-7x-30 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&3&10&-7&-30\\& & -6& -8& \color{black}{30} \\ \hline &\color{blue}{3}&\color{blue}{4}&\color{blue}{-15}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 3x^{3}+10x^{2}-7x-30 }{ x+2 } = \color{blue}{3x^{2}+4x-15} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&10&-7&-30\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 3 }&10&-7&-30\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 3 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&10&-7&-30\\& & \color{blue}{-6} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrr}-2&3&\color{orangered}{ 10 }&-7&-30\\& & \color{orangered}{-6} & & \\ \hline &3&\color{orangered}{4}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 4 } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&10&-7&-30\\& & -6& \color{blue}{-8} & \\ \hline &3&\color{blue}{4}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -15 } $
$$ \begin{array}{c|rrrr}-2&3&10&\color{orangered}{ -7 }&-30\\& & -6& \color{orangered}{-8} & \\ \hline &3&4&\color{orangered}{-15}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -15 \right) } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&10&-7&-30\\& & -6& -8& \color{blue}{30} \\ \hline &3&4&\color{blue}{-15}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -30 } + \color{orangered}{ 30 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-2&3&10&-7&\color{orangered}{ -30 }\\& & -6& -8& \color{orangered}{30} \\ \hline &\color{blue}{3}&\color{blue}{4}&\color{blue}{-15}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}+4x-15 } $ with a remainder of $ \color{red}{ 0 } $.