Divide using synthetic division $$ ( 5x^{3}-2x^{2}-8x+7 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&5&-2&-8&7\\& & -10& 24& \color{black}{-32} \\ \hline &\color{blue}{5}&\color{blue}{-12}&\color{blue}{16}&\color{orangered}{-25} \end{array} $$The solution is:
$$ \frac{ 5x^{3}-2x^{2}-8x+7 }{ x+2 } = \color{blue}{5x^{2}-12x+16} \color{red}{~-~} \frac{ \color{red}{ 25 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&5&-2&-8&7\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 5 }&-2&-8&7\\& & & & \\ \hline &\color{orangered}{5}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 5 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&5&-2&-8&7\\& & \color{blue}{-10} & & \\ \hline &\color{blue}{5}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrrr}-2&5&\color{orangered}{ -2 }&-8&7\\& & \color{orangered}{-10} & & \\ \hline &5&\color{orangered}{-12}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -12 \right) } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&5&-2&-8&7\\& & -10& \color{blue}{24} & \\ \hline &5&\color{blue}{-12}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 24 } = \color{orangered}{ 16 } $
$$ \begin{array}{c|rrrr}-2&5&-2&\color{orangered}{ -8 }&7\\& & -10& \color{orangered}{24} & \\ \hline &5&-12&\color{orangered}{16}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 16 } = \color{blue}{ -32 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&5&-2&-8&7\\& & -10& 24& \color{blue}{-32} \\ \hline &5&-12&\color{blue}{16}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ \left( -32 \right) } = \color{orangered}{ -25 } $
$$ \begin{array}{c|rrrr}-2&5&-2&-8&\color{orangered}{ 7 }\\& & -10& 24& \color{orangered}{-32} \\ \hline &\color{blue}{5}&\color{blue}{-12}&\color{blue}{16}&\color{orangered}{-25} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x^{2}-12x+16 } $ with a remainder of $ \color{red}{ -25 } $.