Divide using synthetic division $$ ( 4x^{3}-3x^{2}+5x-6 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-4&4&-3&5&-6\\& & -16& 76& \color{black}{-324} \\ \hline &\color{blue}{4}&\color{blue}{-19}&\color{blue}{81}&\color{orangered}{-330} \end{array} $$The solution is:
$$ \frac{ 4x^{3}-3x^{2}+5x-6 }{ x+4 } = \color{blue}{4x^{2}-19x+81} \color{red}{~-~} \frac{ \color{red}{ 330 } }{ x+4 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&4&-3&5&-6\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-4&\color{orangered}{ 4 }&-3&5&-6\\& & & & \\ \hline &\color{orangered}{4}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 4 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&4&-3&5&-6\\& & \color{blue}{-16} & & \\ \hline &\color{blue}{4}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -19 } $
$$ \begin{array}{c|rrrr}-4&4&\color{orangered}{ -3 }&5&-6\\& & \color{orangered}{-16} & & \\ \hline &4&\color{orangered}{-19}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -19 \right) } = \color{blue}{ 76 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&4&-3&5&-6\\& & -16& \color{blue}{76} & \\ \hline &4&\color{blue}{-19}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 76 } = \color{orangered}{ 81 } $
$$ \begin{array}{c|rrrr}-4&4&-3&\color{orangered}{ 5 }&-6\\& & -16& \color{orangered}{76} & \\ \hline &4&-19&\color{orangered}{81}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 81 } = \color{blue}{ -324 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&4&-3&5&-6\\& & -16& 76& \color{blue}{-324} \\ \hline &4&-19&\color{blue}{81}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -324 \right) } = \color{orangered}{ -330 } $
$$ \begin{array}{c|rrrr}-4&4&-3&5&\color{orangered}{ -6 }\\& & -16& 76& \color{orangered}{-324} \\ \hline &\color{blue}{4}&\color{blue}{-19}&\color{blue}{81}&\color{orangered}{-330} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{2}-19x+81 } $ with a remainder of $ \color{red}{ -330 } $.