Divide using synthetic division $$ ( 24x^{3}+20x^{2}-18x-12 ) \div ( 6x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-5&24&20&-18&-12\\& & -120& 500& \color{black}{-2410} \\ \hline &\color{blue}{24}&\color{blue}{-100}&\color{blue}{482}&\color{orangered}{-2422} \end{array} $$The solution is:
$$ \frac{ 24x^{3}+20x^{2}-18x-12 }{ x+5 } = \color{blue}{24x^{2}-100x+482} \color{red}{~-~} \frac{ \color{red}{ 2422 } }{ x+5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&24&20&-18&-12\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-5&\color{orangered}{ 24 }&20&-18&-12\\& & & & \\ \hline &\color{orangered}{24}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 24 } = \color{blue}{ -120 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&24&20&-18&-12\\& & \color{blue}{-120} & & \\ \hline &\color{blue}{24}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 20 } + \color{orangered}{ \left( -120 \right) } = \color{orangered}{ -100 } $
$$ \begin{array}{c|rrrr}-5&24&\color{orangered}{ 20 }&-18&-12\\& & \color{orangered}{-120} & & \\ \hline &24&\color{orangered}{-100}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -100 \right) } = \color{blue}{ 500 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&24&20&-18&-12\\& & -120& \color{blue}{500} & \\ \hline &24&\color{blue}{-100}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 500 } = \color{orangered}{ 482 } $
$$ \begin{array}{c|rrrr}-5&24&20&\color{orangered}{ -18 }&-12\\& & -120& \color{orangered}{500} & \\ \hline &24&-100&\color{orangered}{482}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 482 } = \color{blue}{ -2410 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&24&20&-18&-12\\& & -120& 500& \color{blue}{-2410} \\ \hline &24&-100&\color{blue}{482}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ \left( -2410 \right) } = \color{orangered}{ -2422 } $
$$ \begin{array}{c|rrrr}-5&24&20&-18&\color{orangered}{ -12 }\\& & -120& 500& \color{orangered}{-2410} \\ \hline &\color{blue}{24}&\color{blue}{-100}&\color{blue}{482}&\color{orangered}{-2422} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 24x^{2}-100x+482 } $ with a remainder of $ \color{red}{ -2422 } $.