Divide using synthetic division $$ ( 16x^{3}-20x^{2}-4x+15 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-1&16&-20&-4&15\\& & -16& 36& \color{black}{-32} \\ \hline &\color{blue}{16}&\color{blue}{-36}&\color{blue}{32}&\color{orangered}{-17} \end{array} $$The solution is:
$$ \frac{ 16x^{3}-20x^{2}-4x+15 }{ x+1 } = \color{blue}{16x^{2}-36x+32} \color{red}{~-~} \frac{ \color{red}{ 17 } }{ x+1 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 1 = 0 $ ( $ x = \color{blue}{ -1 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&16&-20&-4&15\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-1&\color{orangered}{ 16 }&-20&-4&15\\& & & & \\ \hline &\color{orangered}{16}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 16 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&16&-20&-4&15\\& & \color{blue}{-16} & & \\ \hline &\color{blue}{16}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -36 } $
$$ \begin{array}{c|rrrr}-1&16&\color{orangered}{ -20 }&-4&15\\& & \color{orangered}{-16} & & \\ \hline &16&\color{orangered}{-36}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -36 \right) } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&16&-20&-4&15\\& & -16& \color{blue}{36} & \\ \hline &16&\color{blue}{-36}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 36 } = \color{orangered}{ 32 } $
$$ \begin{array}{c|rrrr}-1&16&-20&\color{orangered}{ -4 }&15\\& & -16& \color{orangered}{36} & \\ \hline &16&-36&\color{orangered}{32}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 32 } = \color{blue}{ -32 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&16&-20&-4&15\\& & -16& 36& \color{blue}{-32} \\ \hline &16&-36&\color{blue}{32}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 15 } + \color{orangered}{ \left( -32 \right) } = \color{orangered}{ -17 } $
$$ \begin{array}{c|rrrr}-1&16&-20&-4&\color{orangered}{ 15 }\\& & -16& 36& \color{orangered}{-32} \\ \hline &\color{blue}{16}&\color{blue}{-36}&\color{blue}{32}&\color{orangered}{-17} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 16x^{2}-36x+32 } $ with a remainder of $ \color{red}{ -17 } $.