Divide using synthetic division $$ ( 3x^{3}+16x^{2}+14x-8 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-4&3&16&14&-8\\& & -12& -16& \color{black}{8} \\ \hline &\color{blue}{3}&\color{blue}{4}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 3x^{3}+16x^{2}+14x-8 }{ x+4 } = \color{blue}{3x^{2}+4x-2} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&3&16&14&-8\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-4&\color{orangered}{ 3 }&16&14&-8\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 3 } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&3&16&14&-8\\& & \color{blue}{-12} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 16 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrr}-4&3&\color{orangered}{ 16 }&14&-8\\& & \color{orangered}{-12} & & \\ \hline &3&\color{orangered}{4}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 4 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&3&16&14&-8\\& & -12& \color{blue}{-16} & \\ \hline &3&\color{blue}{4}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrr}-4&3&16&\color{orangered}{ 14 }&-8\\& & -12& \color{orangered}{-16} & \\ \hline &3&4&\color{orangered}{-2}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&3&16&14&-8\\& & -12& -16& \color{blue}{8} \\ \hline &3&4&\color{blue}{-2}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 8 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-4&3&16&14&\color{orangered}{ -8 }\\& & -12& -16& \color{orangered}{8} \\ \hline &\color{blue}{3}&\color{blue}{4}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}+4x-2 } $ with a remainder of $ \color{red}{ 0 } $.