Divide using synthetic division $$ ( 4x^{3}+5x^{2}-23x-6 ) \div ( 4x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-1&4&5&-23&-6\\& & -4& -1& \color{black}{24} \\ \hline &\color{blue}{4}&\color{blue}{1}&\color{blue}{-24}&\color{orangered}{18} \end{array} $$The solution is:
$$ \frac{ 4x^{3}+5x^{2}-23x-6 }{ x+1 } = \color{blue}{4x^{2}+x-24} ~+~ \frac{ \color{red}{ 18 } }{ x+1 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 1 = 0 $ ( $ x = \color{blue}{ -1 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&4&5&-23&-6\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-1&\color{orangered}{ 4 }&5&-23&-6\\& & & & \\ \hline &\color{orangered}{4}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 4 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&4&5&-23&-6\\& & \color{blue}{-4} & & \\ \hline &\color{blue}{4}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrr}-1&4&\color{orangered}{ 5 }&-23&-6\\& & \color{orangered}{-4} & & \\ \hline &4&\color{orangered}{1}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 1 } = \color{blue}{ -1 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&4&5&-23&-6\\& & -4& \color{blue}{-1} & \\ \hline &4&\color{blue}{1}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -23 } + \color{orangered}{ \left( -1 \right) } = \color{orangered}{ -24 } $
$$ \begin{array}{c|rrrr}-1&4&5&\color{orangered}{ -23 }&-6\\& & -4& \color{orangered}{-1} & \\ \hline &4&1&\color{orangered}{-24}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -24 \right) } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&4&5&-23&-6\\& & -4& -1& \color{blue}{24} \\ \hline &4&1&\color{blue}{-24}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 24 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrr}-1&4&5&-23&\color{orangered}{ -6 }\\& & -4& -1& \color{orangered}{24} \\ \hline &\color{blue}{4}&\color{blue}{1}&\color{blue}{-24}&\color{orangered}{18} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{2}+x-24 } $ with a remainder of $ \color{red}{ 18 } $.