Divide using synthetic division $$ ( 2x^{3}-3x^{2}+4x+3 ) \div ( x+0.5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-\frac{ 1 }{ 2 }&2&-3&4&3\\& & -1& 2& \color{black}{-3} \\ \hline &\color{blue}{2}&\color{blue}{-4}&\color{blue}{6}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 2x^{3}-3x^{2}+4x+3 }{ x+\frac{ 1 }{ 2 } } = \color{blue}{2x^{2}-4x+6} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-\frac{ 1 }{ 2 }}&2&-3&4&3\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-\frac{ 1 }{ 2 }&\color{orangered}{ 2 }&-3&4&3\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -\frac{ 1 }{ 2 } } \cdot \color{blue}{ 2 } = \color{blue}{ -1 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-\frac{ 1 }{ 2 }}&2&-3&4&3\\& & \color{blue}{-1} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -1 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrr}-\frac{ 1 }{ 2 }&2&\color{orangered}{ -3 }&4&3\\& & \color{orangered}{-1} & & \\ \hline &2&\color{orangered}{-4}&& \end{array} $$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}{ \left( -4 \right) } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-\frac{ 1 }{ 2 }}&2&-3&4&3\\& & -1& \color{blue}{2} & \\ \hline &2&\color{blue}{-4}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 2 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrr}-\frac{ 1 }{ 2 }&2&-3&\color{orangered}{ 4 }&3\\& & -1& \color{orangered}{2} & \\ \hline &2&-4&\color{orangered}{6}& \end{array} $$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}{ 6 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-\frac{ 1 }{ 2 }}&2&-3&4&3\\& & -1& 2& \color{blue}{-3} \\ \hline &2&-4&\color{blue}{6}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-\frac{ 1 }{ 2 }&2&-3&4&\color{orangered}{ 3 }\\& & -1& 2& \color{orangered}{-3} \\ \hline &\color{blue}{2}&\color{blue}{-4}&\color{blue}{6}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}-4x+6 } $ with a remainder of $ \color{red}{ 0 } $.