Divide using synthetic division $$ ( -32x^{2}+288 ) \div ( x-0.5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}\frac{ 1 }{ 2 }&-32&0&288\\& & -16& \color{black}{-8} \\ \hline &\color{blue}{-32}&\color{blue}{-16}&\color{orangered}{280} \end{array} $$The solution is:
$$ \frac{ -32x^{2}+288 }{ x-\frac{ 1 }{ 2 } } = \color{blue}{-32x-16} ~+~ \frac{ \color{red}{ 280 } }{ x-\frac{ 1 }{ 2 } } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -\frac{ 1 }{ 2 } = 0 $ ( $ x = \color{blue}{ \frac{ 1 }{ 2 } } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{\frac{ 1 }{ 2 }}&-32&0&288\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}\frac{ 1 }{ 2 }&\color{orangered}{ -32 }&0&288\\& & & \\ \hline &\color{orangered}{-32}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 1 }{ 2 } } \cdot \color{blue}{ \left( -32 \right) } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrr}\color{blue}{\frac{ 1 }{ 2 }}&-32&0&288\\& & \color{blue}{-16} & \\ \hline &\color{blue}{-32}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -16 } $
$$ \begin{array}{c|rrr}\frac{ 1 }{ 2 }&-32&\color{orangered}{ 0 }&288\\& & \color{orangered}{-16} & \\ \hline &-32&\color{orangered}{-16}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 1 }{ 2 } } \cdot \color{blue}{ \left( -16 \right) } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrr}\color{blue}{\frac{ 1 }{ 2 }}&-32&0&288\\& & -16& \color{blue}{-8} \\ \hline &-32&\color{blue}{-16}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 288 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ 280 } $
$$ \begin{array}{c|rrr}\frac{ 1 }{ 2 }&-32&0&\color{orangered}{ 288 }\\& & -16& \color{orangered}{-8} \\ \hline &\color{blue}{-32}&\color{blue}{-16}&\color{orangered}{280} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -32x-16 } $ with a remainder of $ \color{red}{ 280 } $.