Divide using synthetic division $$ ( 16x^{3}+12x^{2}+6x+10 ) \div ( x+0.5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-\frac{ 1 }{ 2 }&16&12&6&10\\& & -8& -2& \color{black}{-2} \\ \hline &\color{blue}{16}&\color{blue}{4}&\color{blue}{4}&\color{orangered}{8} \end{array} $$The solution is:
$$ \frac{ 16x^{3}+12x^{2}+6x+10 }{ x+\frac{ 1 }{ 2 } } = \color{blue}{16x^{2}+4x+4} ~+~ \frac{ \color{red}{ 8 } }{ x+\frac{ 1 }{ 2 } } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-\frac{ 1 }{ 2 }}&16&12&6&10\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-\frac{ 1 }{ 2 }&\color{orangered}{ 16 }&12&6&10\\& & & & \\ \hline &\color{orangered}{16}&&& \end{array} $$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}{ 16 } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-\frac{ 1 }{ 2 }}&16&12&6&10\\& & \color{blue}{-8} & & \\ \hline &\color{blue}{16}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrr}-\frac{ 1 }{ 2 }&16&\color{orangered}{ 12 }&6&10\\& & \color{orangered}{-8} & & \\ \hline &16&\color{orangered}{4}&& \end{array} $$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}{ 4 } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-\frac{ 1 }{ 2 }}&16&12&6&10\\& & -8& \color{blue}{-2} & \\ \hline &16&\color{blue}{4}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrr}-\frac{ 1 }{ 2 }&16&12&\color{orangered}{ 6 }&10\\& & -8& \color{orangered}{-2} & \\ \hline &16&4&\color{orangered}{4}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -\frac{ 1 }{ 2 } } \cdot \color{blue}{ 4 } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-\frac{ 1 }{ 2 }}&16&12&6&10\\& & -8& -2& \color{blue}{-2} \\ \hline &16&4&\color{blue}{4}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrr}-\frac{ 1 }{ 2 }&16&12&6&\color{orangered}{ 10 }\\& & -8& -2& \color{orangered}{-2} \\ \hline &\color{blue}{16}&\color{blue}{4}&\color{blue}{4}&\color{orangered}{8} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 16x^{2}+4x+4 } $ with a remainder of $ \color{red}{ 8 } $.