Divide using synthetic division $$ ( -5x^{2}-12x+7 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-2&-5&-12&7\\& & 10& \color{black}{4} \\ \hline &\color{blue}{-5}&\color{blue}{-2}&\color{orangered}{11} \end{array} $$The solution is:
$$ \frac{ -5x^{2}-12x+7 }{ x+2 } = \color{blue}{-5x-2} ~+~ \frac{ \color{red}{ 11 } }{ x+2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-2}&-5&-12&7\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-2&\color{orangered}{ -5 }&-12&7\\& & & \\ \hline &\color{orangered}{-5}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&-5&-12&7\\& & \color{blue}{10} & \\ \hline &\color{blue}{-5}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ 10 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrr}-2&-5&\color{orangered}{ -12 }&7\\& & \color{orangered}{10} & \\ \hline &-5&\color{orangered}{-2}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&-5&-12&7\\& & 10& \color{blue}{4} \\ \hline &-5&\color{blue}{-2}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ 4 } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrr}-2&-5&-12&\color{orangered}{ 7 }\\& & 10& \color{orangered}{4} \\ \hline &\color{blue}{-5}&\color{blue}{-2}&\color{orangered}{11} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -5x-2 } $ with a remainder of $ \color{red}{ 11 } $.