Divide using synthetic division $$ ( -4x^{2}+6x-7 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-4&-4&6&-7\\& & 16& \color{black}{-88} \\ \hline &\color{blue}{-4}&\color{blue}{22}&\color{orangered}{-95} \end{array} $$The solution is:
$$ \frac{ -4x^{2}+6x-7 }{ x+4 } = \color{blue}{-4x+22} \color{red}{~-~} \frac{ \color{red}{ 95 } }{ x+4 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-4}&-4&6&-7\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-4&\color{orangered}{ -4 }&6&-7\\& & & \\ \hline &\color{orangered}{-4}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&-4&6&-7\\& & \color{blue}{16} & \\ \hline &\color{blue}{-4}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 16 } = \color{orangered}{ 22 } $
$$ \begin{array}{c|rrr}-4&-4&\color{orangered}{ 6 }&-7\\& & \color{orangered}{16} & \\ \hline &-4&\color{orangered}{22}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 22 } = \color{blue}{ -88 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&-4&6&-7\\& & 16& \color{blue}{-88} \\ \hline &-4&\color{blue}{22}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \left( -88 \right) } = \color{orangered}{ -95 } $
$$ \begin{array}{c|rrr}-4&-4&6&\color{orangered}{ -7 }\\& & 16& \color{orangered}{-88} \\ \hline &\color{blue}{-4}&\color{blue}{22}&\color{orangered}{-95} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -4x+22 } $ with a remainder of $ \color{red}{ -95 } $.