Divide using synthetic division $$ ( 4x^{3}-8x^{2}+4x-10 ) \div ( 4x+6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-6&4&-8&4&-10\\& & -24& 192& \color{black}{-1176} \\ \hline &\color{blue}{4}&\color{blue}{-32}&\color{blue}{196}&\color{orangered}{-1186} \end{array} $$The solution is:
$$ \frac{ 4x^{3}-8x^{2}+4x-10 }{ x+6 } = \color{blue}{4x^{2}-32x+196} \color{red}{~-~} \frac{ \color{red}{ 1186 } }{ x+6 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 6 = 0 $ ( $ x = \color{blue}{ -6 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&4&-8&4&-10\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-6&\color{orangered}{ 4 }&-8&4&-10\\& & & & \\ \hline &\color{orangered}{4}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 4 } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&4&-8&4&-10\\& & \color{blue}{-24} & & \\ \hline &\color{blue}{4}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -32 } $
$$ \begin{array}{c|rrrr}-6&4&\color{orangered}{ -8 }&4&-10\\& & \color{orangered}{-24} & & \\ \hline &4&\color{orangered}{-32}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -32 \right) } = \color{blue}{ 192 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&4&-8&4&-10\\& & -24& \color{blue}{192} & \\ \hline &4&\color{blue}{-32}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 192 } = \color{orangered}{ 196 } $
$$ \begin{array}{c|rrrr}-6&4&-8&\color{orangered}{ 4 }&-10\\& & -24& \color{orangered}{192} & \\ \hline &4&-32&\color{orangered}{196}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 196 } = \color{blue}{ -1176 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&4&-8&4&-10\\& & -24& 192& \color{blue}{-1176} \\ \hline &4&-32&\color{blue}{196}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ \left( -1176 \right) } = \color{orangered}{ -1186 } $
$$ \begin{array}{c|rrrr}-6&4&-8&4&\color{orangered}{ -10 }\\& & -24& 192& \color{orangered}{-1176} \\ \hline &\color{blue}{4}&\color{blue}{-32}&\color{blue}{196}&\color{orangered}{-1186} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{2}-32x+196 } $ with a remainder of $ \color{red}{ -1186 } $.