Divide using synthetic division $$ ( 4x^{3}-105x+26 ) \div ( 4x+9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-9&4&0&-105&26\\& & -36& 324& \color{black}{-1971} \\ \hline &\color{blue}{4}&\color{blue}{-36}&\color{blue}{219}&\color{orangered}{-1945} \end{array} $$The solution is:
$$ \frac{ 4x^{3}-105x+26 }{ x+9 } = \color{blue}{4x^{2}-36x+219} \color{red}{~-~} \frac{ \color{red}{ 1945 } }{ x+9 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 9 = 0 $ ( $ x = \color{blue}{ -9 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&4&0&-105&26\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-9&\color{orangered}{ 4 }&0&-105&26\\& & & & \\ \hline &\color{orangered}{4}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ 4 } = \color{blue}{ -36 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&4&0&-105&26\\& & \color{blue}{-36} & & \\ \hline &\color{blue}{4}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -36 \right) } = \color{orangered}{ -36 } $
$$ \begin{array}{c|rrrr}-9&4&\color{orangered}{ 0 }&-105&26\\& & \color{orangered}{-36} & & \\ \hline &4&\color{orangered}{-36}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ \left( -36 \right) } = \color{blue}{ 324 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&4&0&-105&26\\& & -36& \color{blue}{324} & \\ \hline &4&\color{blue}{-36}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -105 } + \color{orangered}{ 324 } = \color{orangered}{ 219 } $
$$ \begin{array}{c|rrrr}-9&4&0&\color{orangered}{ -105 }&26\\& & -36& \color{orangered}{324} & \\ \hline &4&-36&\color{orangered}{219}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ 219 } = \color{blue}{ -1971 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&4&0&-105&26\\& & -36& 324& \color{blue}{-1971} \\ \hline &4&-36&\color{blue}{219}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 26 } + \color{orangered}{ \left( -1971 \right) } = \color{orangered}{ -1945 } $
$$ \begin{array}{c|rrrr}-9&4&0&-105&\color{orangered}{ 26 }\\& & -36& 324& \color{orangered}{-1971} \\ \hline &\color{blue}{4}&\color{blue}{-36}&\color{blue}{219}&\color{orangered}{-1945} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{2}-36x+219 } $ with a remainder of $ \color{red}{ -1945 } $.