Divide using synthetic division $$ ( -12x^{3}-5x^{2}+17x+10 ) \div ( 3x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&-12&-5&17&10\\& & 24& -38& \color{black}{42} \\ \hline &\color{blue}{-12}&\color{blue}{19}&\color{blue}{-21}&\color{orangered}{52} \end{array} $$The solution is:
$$ \frac{ -12x^{3}-5x^{2}+17x+10 }{ x+2 } = \color{blue}{-12x^{2}+19x-21} ~+~ \frac{ \color{red}{ 52 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-12&-5&17&10\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ -12 }&-5&17&10\\& & & & \\ \hline &\color{orangered}{-12}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -12 \right) } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-12&-5&17&10\\& & \color{blue}{24} & & \\ \hline &\color{blue}{-12}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 24 } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrrr}-2&-12&\color{orangered}{ -5 }&17&10\\& & \color{orangered}{24} & & \\ \hline &-12&\color{orangered}{19}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 19 } = \color{blue}{ -38 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-12&-5&17&10\\& & 24& \color{blue}{-38} & \\ \hline &-12&\color{blue}{19}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ \left( -38 \right) } = \color{orangered}{ -21 } $
$$ \begin{array}{c|rrrr}-2&-12&-5&\color{orangered}{ 17 }&10\\& & 24& \color{orangered}{-38} & \\ \hline &-12&19&\color{orangered}{-21}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -21 \right) } = \color{blue}{ 42 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-12&-5&17&10\\& & 24& -38& \color{blue}{42} \\ \hline &-12&19&\color{blue}{-21}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ 42 } = \color{orangered}{ 52 } $
$$ \begin{array}{c|rrrr}-2&-12&-5&17&\color{orangered}{ 10 }\\& & 24& -38& \color{orangered}{42} \\ \hline &\color{blue}{-12}&\color{blue}{19}&\color{blue}{-21}&\color{orangered}{52} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -12x^{2}+19x-21 } $ with a remainder of $ \color{red}{ 52 } $.