Divide using synthetic division $$ ( 20x^{3}+44x^{2}-17x-5 ) \div ( x+2.5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-\frac{ 5 }{ 2 }&20&44&-17&-5\\& & -50& 15& \color{black}{5} \\ \hline &\color{blue}{20}&\color{blue}{-6}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 20x^{3}+44x^{2}-17x-5 }{ x+\frac{ 5 }{ 2 } } = \color{blue}{20x^{2}-6x-2} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + \frac{ 5 }{ 2 } = 0 $ ( $ x = \color{blue}{ -\frac{ 5 }{ 2 } } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-\frac{ 5 }{ 2 }}&20&44&-17&-5\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-\frac{ 5 }{ 2 }&\color{orangered}{ 20 }&44&-17&-5\\& & & & \\ \hline &\color{orangered}{20}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -\frac{ 5 }{ 2 } } \cdot \color{blue}{ 20 } = \color{blue}{ -50 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-\frac{ 5 }{ 2 }}&20&44&-17&-5\\& & \color{blue}{-50} & & \\ \hline &\color{blue}{20}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 44 } + \color{orangered}{ \left( -50 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrr}-\frac{ 5 }{ 2 }&20&\color{orangered}{ 44 }&-17&-5\\& & \color{orangered}{-50} & & \\ \hline &20&\color{orangered}{-6}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -\frac{ 5 }{ 2 } } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 15 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-\frac{ 5 }{ 2 }}&20&44&-17&-5\\& & -50& \color{blue}{15} & \\ \hline &20&\color{blue}{-6}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 15 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrr}-\frac{ 5 }{ 2 }&20&44&\color{orangered}{ -17 }&-5\\& & -50& \color{orangered}{15} & \\ \hline &20&-6&\color{orangered}{-2}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -\frac{ 5 }{ 2 } } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 5 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-\frac{ 5 }{ 2 }}&20&44&-17&-5\\& & -50& 15& \color{blue}{5} \\ \hline &20&-6&\color{blue}{-2}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 5 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-\frac{ 5 }{ 2 }&20&44&-17&\color{orangered}{ -5 }\\& & -50& 15& \color{orangered}{5} \\ \hline &\color{blue}{20}&\color{blue}{-6}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 20x^{2}-6x-2 } $ with a remainder of $ \color{red}{ 0 } $.