Divide using synthetic division $$ ( 16x^{2}+53x+21 ) \div ( 2x+7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-7&16&53&21\\& & -112& \color{black}{413} \\ \hline &\color{blue}{16}&\color{blue}{-59}&\color{orangered}{434} \end{array} $$The solution is:
$$ \frac{ 16x^{2}+53x+21 }{ x+7 } = \color{blue}{16x-59} ~+~ \frac{ \color{red}{ 434 } }{ x+7 } $$Explanation
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.
$$ \begin{array}{c|rrr}\color{blue}{-7}&16&53&21\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-7&\color{orangered}{ 16 }&53&21\\& & & \\ \hline &\color{orangered}{16}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 16 } = \color{blue}{ -112 } $.
$$ \begin{array}{c|rrr}\color{blue}{-7}&16&53&21\\& & \color{blue}{-112} & \\ \hline &\color{blue}{16}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 53 } + \color{orangered}{ \left( -112 \right) } = \color{orangered}{ -59 } $
$$ \begin{array}{c|rrr}-7&16&\color{orangered}{ 53 }&21\\& & \color{orangered}{-112} & \\ \hline &16&\color{orangered}{-59}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ \left( -59 \right) } = \color{blue}{ 413 } $.
$$ \begin{array}{c|rrr}\color{blue}{-7}&16&53&21\\& & -112& \color{blue}{413} \\ \hline &16&\color{blue}{-59}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ 413 } = \color{orangered}{ 434 } $
$$ \begin{array}{c|rrr}-7&16&53&\color{orangered}{ 21 }\\& & -112& \color{orangered}{413} \\ \hline &\color{blue}{16}&\color{blue}{-59}&\color{orangered}{434} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 16x-59 } $ with a remainder of $ \color{red}{ 434 } $.