Divide using synthetic division $$ ( 2x^{3}-5x^{2}-13x-15 ) \div ( 2x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-5&2&-5&-13&-15\\& & -10& 75& \color{black}{-310} \\ \hline &\color{blue}{2}&\color{blue}{-15}&\color{blue}{62}&\color{orangered}{-325} \end{array} $$The solution is:
$$ \frac{ 2x^{3}-5x^{2}-13x-15 }{ x+5 } = \color{blue}{2x^{2}-15x+62} \color{red}{~-~} \frac{ \color{red}{ 325 } }{ x+5 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&2&-5&-13&-15\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-5&\color{orangered}{ 2 }&-5&-13&-15\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&2&-5&-13&-15\\& & \color{blue}{-10} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -15 } $
$$ \begin{array}{c|rrrr}-5&2&\color{orangered}{ -5 }&-13&-15\\& & \color{orangered}{-10} & & \\ \hline &2&\color{orangered}{-15}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -15 \right) } = \color{blue}{ 75 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&2&-5&-13&-15\\& & -10& \color{blue}{75} & \\ \hline &2&\color{blue}{-15}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 75 } = \color{orangered}{ 62 } $
$$ \begin{array}{c|rrrr}-5&2&-5&\color{orangered}{ -13 }&-15\\& & -10& \color{orangered}{75} & \\ \hline &2&-15&\color{orangered}{62}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 62 } = \color{blue}{ -310 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&2&-5&-13&-15\\& & -10& 75& \color{blue}{-310} \\ \hline &2&-15&\color{blue}{62}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ \left( -310 \right) } = \color{orangered}{ -325 } $
$$ \begin{array}{c|rrrr}-5&2&-5&-13&\color{orangered}{ -15 }\\& & -10& 75& \color{orangered}{-310} \\ \hline &\color{blue}{2}&\color{blue}{-15}&\color{blue}{62}&\color{orangered}{-325} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}-15x+62 } $ with a remainder of $ \color{red}{ -325 } $.