Divide using synthetic division $$ ( -4x^{4}-7x^{2}-7x+2 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-2&-4&0&-7&-7&2\\& & 8& -16& 46& \color{black}{-78} \\ \hline &\color{blue}{-4}&\color{blue}{8}&\color{blue}{-23}&\color{blue}{39}&\color{orangered}{-76} \end{array} $$The solution is:
$$ \frac{ -4x^{4}-7x^{2}-7x+2 }{ x+2 } = \color{blue}{-4x^{3}+8x^{2}-23x+39} \color{red}{~-~} \frac{ \color{red}{ 76 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&-4&0&-7&-7&2\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-2&\color{orangered}{ -4 }&0&-7&-7&2\\& & & & & \\ \hline &\color{orangered}{-4}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&-4&0&-7&-7&2\\& & \color{blue}{8} & & & \\ \hline &\color{blue}{-4}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 8 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrr}-2&-4&\color{orangered}{ 0 }&-7&-7&2\\& & \color{orangered}{8} & & & \\ \hline &-4&\color{orangered}{8}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 8 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&-4&0&-7&-7&2\\& & 8& \color{blue}{-16} & & \\ \hline &-4&\color{blue}{8}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -23 } $
$$ \begin{array}{c|rrrrr}-2&-4&0&\color{orangered}{ -7 }&-7&2\\& & 8& \color{orangered}{-16} & & \\ \hline &-4&8&\color{orangered}{-23}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -23 \right) } = \color{blue}{ 46 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&-4&0&-7&-7&2\\& & 8& -16& \color{blue}{46} & \\ \hline &-4&8&\color{blue}{-23}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 46 } = \color{orangered}{ 39 } $
$$ \begin{array}{c|rrrrr}-2&-4&0&-7&\color{orangered}{ -7 }&2\\& & 8& -16& \color{orangered}{46} & \\ \hline &-4&8&-23&\color{orangered}{39}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 39 } = \color{blue}{ -78 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&-4&0&-7&-7&2\\& & 8& -16& 46& \color{blue}{-78} \\ \hline &-4&8&-23&\color{blue}{39}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -78 \right) } = \color{orangered}{ -76 } $
$$ \begin{array}{c|rrrrr}-2&-4&0&-7&-7&\color{orangered}{ 2 }\\& & 8& -16& 46& \color{orangered}{-78} \\ \hline &\color{blue}{-4}&\color{blue}{8}&\color{blue}{-23}&\color{blue}{39}&\color{orangered}{-76} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -4x^{3}+8x^{2}-23x+39 } $ with a remainder of $ \color{red}{ -76 } $.