Divide using synthetic division $$ ( 3x^{4}+6x^{3}-15x^{2}+32x-25 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-4&3&6&-15&32&-25\\& & -12& 24& -36& \color{black}{16} \\ \hline &\color{blue}{3}&\color{blue}{-6}&\color{blue}{9}&\color{blue}{-4}&\color{orangered}{-9} \end{array} $$The solution is:
$$ \frac{ 3x^{4}+6x^{3}-15x^{2}+32x-25 }{ x+4 } = \color{blue}{3x^{3}-6x^{2}+9x-4} \color{red}{~-~} \frac{ \color{red}{ 9 } }{ x+4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&3&6&-15&32&-25\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-4&\color{orangered}{ 3 }&6&-15&32&-25\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 3 } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&3&6&-15&32&-25\\& & \color{blue}{-12} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrr}-4&3&\color{orangered}{ 6 }&-15&32&-25\\& & \color{orangered}{-12} & & & \\ \hline &3&\color{orangered}{-6}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&3&6&-15&32&-25\\& & -12& \color{blue}{24} & & \\ \hline &3&\color{blue}{-6}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ 24 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrrr}-4&3&6&\color{orangered}{ -15 }&32&-25\\& & -12& \color{orangered}{24} & & \\ \hline &3&-6&\color{orangered}{9}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 9 } = \color{blue}{ -36 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&3&6&-15&32&-25\\& & -12& 24& \color{blue}{-36} & \\ \hline &3&-6&\color{blue}{9}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 32 } + \color{orangered}{ \left( -36 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrr}-4&3&6&-15&\color{orangered}{ 32 }&-25\\& & -12& 24& \color{orangered}{-36} & \\ \hline &3&-6&9&\color{orangered}{-4}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&3&6&-15&32&-25\\& & -12& 24& -36& \color{blue}{16} \\ \hline &3&-6&9&\color{blue}{-4}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -25 } + \color{orangered}{ 16 } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrr}-4&3&6&-15&32&\color{orangered}{ -25 }\\& & -12& 24& -36& \color{orangered}{16} \\ \hline &\color{blue}{3}&\color{blue}{-6}&\color{blue}{9}&\color{blue}{-4}&\color{orangered}{-9} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{3}-6x^{2}+9x-4 } $ with a remainder of $ \color{red}{ -9 } $.