Divide using synthetic division $$ ( 36x^{3}-19x+2 ) \div ( 3x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&36&0&-19&2\\& & -72& 144& \color{black}{-250} \\ \hline &\color{blue}{36}&\color{blue}{-72}&\color{blue}{125}&\color{orangered}{-248} \end{array} $$The solution is:
$$ \frac{ 36x^{3}-19x+2 }{ x+2 } = \color{blue}{36x^{2}-72x+125} \color{red}{~-~} \frac{ \color{red}{ 248 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&36&0&-19&2\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 36 }&0&-19&2\\& & & & \\ \hline &\color{orangered}{36}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 36 } = \color{blue}{ -72 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&36&0&-19&2\\& & \color{blue}{-72} & & \\ \hline &\color{blue}{36}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -72 \right) } = \color{orangered}{ -72 } $
$$ \begin{array}{c|rrrr}-2&36&\color{orangered}{ 0 }&-19&2\\& & \color{orangered}{-72} & & \\ \hline &36&\color{orangered}{-72}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -72 \right) } = \color{blue}{ 144 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&36&0&-19&2\\& & -72& \color{blue}{144} & \\ \hline &36&\color{blue}{-72}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -19 } + \color{orangered}{ 144 } = \color{orangered}{ 125 } $
$$ \begin{array}{c|rrrr}-2&36&0&\color{orangered}{ -19 }&2\\& & -72& \color{orangered}{144} & \\ \hline &36&-72&\color{orangered}{125}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 125 } = \color{blue}{ -250 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&36&0&-19&2\\& & -72& 144& \color{blue}{-250} \\ \hline &36&-72&\color{blue}{125}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -250 \right) } = \color{orangered}{ -248 } $
$$ \begin{array}{c|rrrr}-2&36&0&-19&\color{orangered}{ 2 }\\& & -72& 144& \color{orangered}{-250} \\ \hline &\color{blue}{36}&\color{blue}{-72}&\color{blue}{125}&\color{orangered}{-248} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 36x^{2}-72x+125 } $ with a remainder of $ \color{red}{ -248 } $.