Divide using synthetic division $$ ( 9x^{4}-4x^{2}-5 ) \div ( 3x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-2&9&0&-4&0&-5\\& & -18& 36& -64& \color{black}{128} \\ \hline &\color{blue}{9}&\color{blue}{-18}&\color{blue}{32}&\color{blue}{-64}&\color{orangered}{123} \end{array} $$The solution is:
$$ \frac{ 9x^{4}-4x^{2}-5 }{ x+2 } = \color{blue}{9x^{3}-18x^{2}+32x-64} ~+~ \frac{ \color{red}{ 123 } }{ x+2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&9&0&-4&0&-5\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-2&\color{orangered}{ 9 }&0&-4&0&-5\\& & & & & \\ \hline &\color{orangered}{9}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 9 } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&9&0&-4&0&-5\\& & \color{blue}{-18} & & & \\ \hline &\color{blue}{9}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -18 } $
$$ \begin{array}{c|rrrrr}-2&9&\color{orangered}{ 0 }&-4&0&-5\\& & \color{orangered}{-18} & & & \\ \hline &9&\color{orangered}{-18}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -18 \right) } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&9&0&-4&0&-5\\& & -18& \color{blue}{36} & & \\ \hline &9&\color{blue}{-18}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 36 } = \color{orangered}{ 32 } $
$$ \begin{array}{c|rrrrr}-2&9&0&\color{orangered}{ -4 }&0&-5\\& & -18& \color{orangered}{36} & & \\ \hline &9&-18&\color{orangered}{32}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 32 } = \color{blue}{ -64 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&9&0&-4&0&-5\\& & -18& 36& \color{blue}{-64} & \\ \hline &9&-18&\color{blue}{32}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -64 \right) } = \color{orangered}{ -64 } $
$$ \begin{array}{c|rrrrr}-2&9&0&-4&\color{orangered}{ 0 }&-5\\& & -18& 36& \color{orangered}{-64} & \\ \hline &9&-18&32&\color{orangered}{-64}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -64 \right) } = \color{blue}{ 128 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&9&0&-4&0&-5\\& & -18& 36& -64& \color{blue}{128} \\ \hline &9&-18&32&\color{blue}{-64}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 128 } = \color{orangered}{ 123 } $
$$ \begin{array}{c|rrrrr}-2&9&0&-4&0&\color{orangered}{ -5 }\\& & -18& 36& -64& \color{orangered}{128} \\ \hline &\color{blue}{9}&\color{blue}{-18}&\color{blue}{32}&\color{blue}{-64}&\color{orangered}{123} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x^{3}-18x^{2}+32x-64 } $ with a remainder of $ \color{red}{ 123 } $.