Divide using synthetic division $$ ( 2x^{3}+17x^{2}+38x+15 ) \div ( 2x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-5&2&17&38&15\\& & -10& -35& \color{black}{-15} \\ \hline &\color{blue}{2}&\color{blue}{7}&\color{blue}{3}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 2x^{3}+17x^{2}+38x+15 }{ x+5 } = \color{blue}{2x^{2}+7x+3} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&2&17&38&15\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-5&\color{orangered}{ 2 }&17&38&15\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 2 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&2&17&38&15\\& & \color{blue}{-10} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrr}-5&2&\color{orangered}{ 17 }&38&15\\& & \color{orangered}{-10} & & \\ \hline &2&\color{orangered}{7}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 7 } = \color{blue}{ -35 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&2&17&38&15\\& & -10& \color{blue}{-35} & \\ \hline &2&\color{blue}{7}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 38 } + \color{orangered}{ \left( -35 \right) } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrr}-5&2&17&\color{orangered}{ 38 }&15\\& & -10& \color{orangered}{-35} & \\ \hline &2&7&\color{orangered}{3}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 3 } = \color{blue}{ -15 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&2&17&38&15\\& & -10& -35& \color{blue}{-15} \\ \hline &2&7&\color{blue}{3}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 15 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-5&2&17&38&\color{orangered}{ 15 }\\& & -10& -35& \color{orangered}{-15} \\ \hline &\color{blue}{2}&\color{blue}{7}&\color{blue}{3}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}+7x+3 } $ with a remainder of $ \color{red}{ 0 } $.