Divide using synthetic division $$ ( 4x^{4}-5x^{2}+1 ) \div ( x-0.5 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}\dfrac{ 1 }{ 2 }&4&0&-5&0&1\\& & 2& 1& -2& \color{black}{-1} \\ \hline &\color{blue}{4}&\color{blue}{2}&\color{blue}{-4}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$The solution is:
$$ \dfrac{ 4x^{4}-5x^{2}+1 }{ x-\dfrac{ 1 }{ 2 } } = \color{blue}{4x^{3}+2x^{2}-4x-2} $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{\frac{ 1 }{ 2 }}&4&0&-5&0&1\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}\frac{ 1 }{ 2 }&\color{orangered}{ 4 }&0&-5&0&1\\& & & & & \\ \hline &\color{orangered}{4}&&&& \end{array} $$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}{ 4 } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{\frac{ 1 }{ 2 }}&4&0&-5&0&1\\& & \color{blue}{2} & & & \\ \hline &\color{blue}{4}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 2 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrr}\frac{ 1 }{ 2 }&4&\color{orangered}{ 0 }&-5&0&1\\& & \color{orangered}{2} & & & \\ \hline &4&\color{orangered}{2}&&& \end{array} $$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}{ 2 } = \color{blue}{ 1 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{\frac{ 1 }{ 2 }}&4&0&-5&0&1\\& & 2& \color{blue}{1} & & \\ \hline &4&\color{blue}{2}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 1 } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrr}\frac{ 1 }{ 2 }&4&0&\color{orangered}{ -5 }&0&1\\& & 2& \color{orangered}{1} & & \\ \hline &4&2&\color{orangered}{-4}&& \end{array} $$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}{ \left( -4 \right) } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{\frac{ 1 }{ 2 }}&4&0&-5&0&1\\& & 2& 1& \color{blue}{-2} & \\ \hline &4&2&\color{blue}{-4}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrr}\frac{ 1 }{ 2 }&4&0&-5&\color{orangered}{ 0 }&1\\& & 2& 1& \color{orangered}{-2} & \\ \hline &4&2&-4&\color{orangered}{-2}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ \frac{ 1 }{ 2 } } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -1 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{\frac{ 1 }{ 2 }}&4&0&-5&0&1\\& & 2& 1& -2& \color{blue}{-1} \\ \hline &4&2&-4&\color{blue}{-2}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -1 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}\frac{ 1 }{ 2 }&4&0&-5&0&\color{orangered}{ 1 }\\& & 2& 1& -2& \color{orangered}{-1} \\ \hline &\color{blue}{4}&\color{blue}{2}&\color{blue}{-4}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{3}+2x^{2}-4x-2 } $ with a remainder of $ \color{red}{ 0 } $.