Divide using synthetic division $$ ( 7x^{2}+23x-20 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-4&7&23&-20\\& & -28& \color{black}{20} \\ \hline &\color{blue}{7}&\color{blue}{-5}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 7x^{2}+23x-20 }{ x+4 } = \color{blue}{7x-5} $$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.
$$ \begin{array}{c|rrr}\color{blue}{-4}&7&23&-20\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-4&\color{orangered}{ 7 }&23&-20\\& & & \\ \hline &\color{orangered}{7}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 7 } = \color{blue}{ -28 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&7&23&-20\\& & \color{blue}{-28} & \\ \hline &\color{blue}{7}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 23 } + \color{orangered}{ \left( -28 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrr}-4&7&\color{orangered}{ 23 }&-20\\& & \color{orangered}{-28} & \\ \hline &7&\color{orangered}{-5}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 20 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&7&23&-20\\& & -28& \color{blue}{20} \\ \hline &7&\color{blue}{-5}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 20 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrr}-4&7&23&\color{orangered}{ -20 }\\& & -28& \color{orangered}{20} \\ \hline &\color{blue}{7}&\color{blue}{-5}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 7x-5 } $ with a remainder of $ \color{red}{ 0 } $.