Divide using synthetic division $$ ( -3x^{2}-10x+24 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-3&-3&-10&24\\& & 9& \color{black}{3} \\ \hline &\color{blue}{-3}&\color{blue}{-1}&\color{orangered}{27} \end{array} $$The solution is:
$$ \frac{ -3x^{2}-10x+24 }{ x+3 } = \color{blue}{-3x-1} ~+~ \frac{ \color{red}{ 27 } }{ x+3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-3}&-3&-10&24\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-3&\color{orangered}{ -3 }&-10&24\\& & & \\ \hline &\color{orangered}{-3}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrr}\color{blue}{-3}&-3&-10&24\\& & \color{blue}{9} & \\ \hline &\color{blue}{-3}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 9 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrr}-3&-3&\color{orangered}{ -10 }&24\\& & \color{orangered}{9} & \\ \hline &-3&\color{orangered}{-1}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 3 } $.
$$ \begin{array}{c|rrr}\color{blue}{-3}&-3&-10&24\\& & 9& \color{blue}{3} \\ \hline &-3&\color{blue}{-1}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ 3 } = \color{orangered}{ 27 } $
$$ \begin{array}{c|rrr}-3&-3&-10&\color{orangered}{ 24 }\\& & 9& \color{orangered}{3} \\ \hline &\color{blue}{-3}&\color{blue}{-1}&\color{orangered}{27} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -3x-1 } $ with a remainder of $ \color{red}{ 27 } $.