Divide using synthetic division $$ ( x^{3}+8x^{2}+12x-10 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-5&1&8&12&-10\\& & -5& -15& \color{black}{15} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{-3}&\color{orangered}{5} \end{array} $$The solution is:
$$ \frac{ x^{3}+8x^{2}+12x-10 }{ x+5 } = \color{blue}{x^{2}+3x-3} ~+~ \frac{ \color{red}{ 5 } }{ x+5 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&1&8&12&-10\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-5&\color{orangered}{ 1 }&8&12&-10\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 1 } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&1&8&12&-10\\& & \color{blue}{-5} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrr}-5&1&\color{orangered}{ 8 }&12&-10\\& & \color{orangered}{-5} & & \\ \hline &1&\color{orangered}{3}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 3 } = \color{blue}{ -15 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&1&8&12&-10\\& & -5& \color{blue}{-15} & \\ \hline &1&\color{blue}{3}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrr}-5&1&8&\color{orangered}{ 12 }&-10\\& & -5& \color{orangered}{-15} & \\ \hline &1&3&\color{orangered}{-3}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 15 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&1&8&12&-10\\& & -5& -15& \color{blue}{15} \\ \hline &1&3&\color{blue}{-3}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 15 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrr}-5&1&8&12&\color{orangered}{ -10 }\\& & -5& -15& \color{orangered}{15} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{-3}&\color{orangered}{5} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+3x-3 } $ with a remainder of $ \color{red}{ 5 } $.