Divide using synthetic division $$ ( 4x^{3}+19x^{2}-61x+14 ) \div ( x+7 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrr}-7&4&19&-61&14\\& & -28& 63& \color{black}{-14} \\ \hline &\color{blue}{4}&\color{blue}{-9}&\color{blue}{2}&\color{orangered}{0} \end{array} $$The solution is:
$$ \dfrac{ 4x^{3}+19x^{2}-61x+14 }{ x+7 } = \color{blue}{4x^{2}-9x+2} $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 7 = 0 $ ( $ x = \color{blue}{ -7 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&4&19&-61&14\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-7&\color{orangered}{ 4 }&19&-61&14\\& & & & \\ \hline &\color{orangered}{4}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 4 } = \color{blue}{ -28 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&4&19&-61&14\\& & \color{blue}{-28} & & \\ \hline &\color{blue}{4}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 19 } + \color{orangered}{ \left( -28 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrr}-7&4&\color{orangered}{ 19 }&-61&14\\& & \color{orangered}{-28} & & \\ \hline &4&\color{orangered}{-9}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 63 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&4&19&-61&14\\& & -28& \color{blue}{63} & \\ \hline &4&\color{blue}{-9}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -61 } + \color{orangered}{ 63 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrr}-7&4&19&\color{orangered}{ -61 }&14\\& & -28& \color{orangered}{63} & \\ \hline &4&-9&\color{orangered}{2}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 2 } = \color{blue}{ -14 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&4&19&-61&14\\& & -28& 63& \color{blue}{-14} \\ \hline &4&-9&\color{blue}{2}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ \left( -14 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-7&4&19&-61&\color{orangered}{ 14 }\\& & -28& 63& \color{orangered}{-14} \\ \hline &\color{blue}{4}&\color{blue}{-9}&\color{blue}{2}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{2}-9x+2 } $ with a remainder of $ \color{red}{ 0 } $.