Divide using synthetic division $$ ( 3x^{2}+x-12 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-5&3&1&-12\\& & -15& \color{black}{70} \\ \hline &\color{blue}{3}&\color{blue}{-14}&\color{orangered}{58} \end{array} $$The solution is:
$$ \frac{ 3x^{2}+x-12 }{ x+5 } = \color{blue}{3x-14} ~+~ \frac{ \color{red}{ 58 } }{ x+5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-5}&3&1&-12\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-5&\color{orangered}{ 3 }&1&-12\\& & & \\ \hline &\color{orangered}{3}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 3 } = \color{blue}{ -15 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&3&1&-12\\& & \color{blue}{-15} & \\ \hline &\color{blue}{3}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ -14 } $
$$ \begin{array}{c|rrr}-5&3&\color{orangered}{ 1 }&-12\\& & \color{orangered}{-15} & \\ \hline &3&\color{orangered}{-14}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -14 \right) } = \color{blue}{ 70 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&3&1&-12\\& & -15& \color{blue}{70} \\ \hline &3&\color{blue}{-14}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ 70 } = \color{orangered}{ 58 } $
$$ \begin{array}{c|rrr}-5&3&1&\color{orangered}{ -12 }\\& & -15& \color{orangered}{70} \\ \hline &\color{blue}{3}&\color{blue}{-14}&\color{orangered}{58} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x-14 } $ with a remainder of $ \color{red}{ 58 } $.