Divide using synthetic division $$ ( 3x^{3}-5x^{2}+4x+4 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-4&3&-5&4&4\\& & -12& 68& \color{black}{-288} \\ \hline &\color{blue}{3}&\color{blue}{-17}&\color{blue}{72}&\color{orangered}{-284} \end{array} $$The solution is:
$$ \frac{ 3x^{3}-5x^{2}+4x+4 }{ x+4 } = \color{blue}{3x^{2}-17x+72} \color{red}{~-~} \frac{ \color{red}{ 284 } }{ x+4 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&3&-5&4&4\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-4&\color{orangered}{ 3 }&-5&4&4\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&3&-5&4&4\\& & \color{blue}{-12} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -17 } $
$$ \begin{array}{c|rrrr}-4&3&\color{orangered}{ -5 }&4&4\\& & \color{orangered}{-12} & & \\ \hline &3&\color{orangered}{-17}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -17 \right) } = \color{blue}{ 68 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&3&-5&4&4\\& & -12& \color{blue}{68} & \\ \hline &3&\color{blue}{-17}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 68 } = \color{orangered}{ 72 } $
$$ \begin{array}{c|rrrr}-4&3&-5&\color{orangered}{ 4 }&4\\& & -12& \color{orangered}{68} & \\ \hline &3&-17&\color{orangered}{72}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 72 } = \color{blue}{ -288 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&3&-5&4&4\\& & -12& 68& \color{blue}{-288} \\ \hline &3&-17&\color{blue}{72}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -288 \right) } = \color{orangered}{ -284 } $
$$ \begin{array}{c|rrrr}-4&3&-5&4&\color{orangered}{ 4 }\\& & -12& 68& \color{orangered}{-288} \\ \hline &\color{blue}{3}&\color{blue}{-17}&\color{blue}{72}&\color{orangered}{-284} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}-17x+72 } $ with a remainder of $ \color{red}{ -284 } $.