Divide using synthetic division $$ ( 3x^{3}+13x^{2}+15x+2 ) \div ( 2x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&3&13&15&2\\& & -6& -14& \color{black}{-2} \\ \hline &\color{blue}{3}&\color{blue}{7}&\color{blue}{1}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 3x^{3}+13x^{2}+15x+2 }{ x+2 } = \color{blue}{3x^{2}+7x+1} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&13&15&2\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 3 }&13&15&2\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 3 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&13&15&2\\& & \color{blue}{-6} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 13 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrr}-2&3&\color{orangered}{ 13 }&15&2\\& & \color{orangered}{-6} & & \\ \hline &3&\color{orangered}{7}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 7 } = \color{blue}{ -14 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&13&15&2\\& & -6& \color{blue}{-14} & \\ \hline &3&\color{blue}{7}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 15 } + \color{orangered}{ \left( -14 \right) } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrr}-2&3&13&\color{orangered}{ 15 }&2\\& & -6& \color{orangered}{-14} & \\ \hline &3&7&\color{orangered}{1}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 1 } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&13&15&2\\& & -6& -14& \color{blue}{-2} \\ \hline &3&7&\color{blue}{1}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-2&3&13&15&\color{orangered}{ 2 }\\& & -6& -14& \color{orangered}{-2} \\ \hline &\color{blue}{3}&\color{blue}{7}&\color{blue}{1}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}+7x+1 } $ with a remainder of $ \color{red}{ 0 } $.