Divide using synthetic division $$ ( 5x^{2}+8x-48 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-4&5&8&-48\\& & -20& \color{black}{48} \\ \hline &\color{blue}{5}&\color{blue}{-12}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 5x^{2}+8x-48 }{ x+4 } = \color{blue}{5x-12} $$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.
$$ \begin{array}{c|rrr}\color{blue}{-4}&5&8&-48\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-4&\color{orangered}{ 5 }&8&-48\\& & & \\ \hline &\color{orangered}{5}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 5 } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&5&8&-48\\& & \color{blue}{-20} & \\ \hline &\color{blue}{5}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrr}-4&5&\color{orangered}{ 8 }&-48\\& & \color{orangered}{-20} & \\ \hline &5&\color{orangered}{-12}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -12 \right) } = \color{blue}{ 48 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&5&8&-48\\& & -20& \color{blue}{48} \\ \hline &5&\color{blue}{-12}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -48 } + \color{orangered}{ 48 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrr}-4&5&8&\color{orangered}{ -48 }\\& & -20& \color{orangered}{48} \\ \hline &\color{blue}{5}&\color{blue}{-12}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x-12 } $ with a remainder of $ \color{red}{ 0 } $.