Check if $ x+3 $ is a factor of $ 22x^{2}-7x-1 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-3&22&-7&-1\\& & -66& \color{black}{219} \\ \hline &\color{blue}{22}&\color{blue}{-73}&\color{orangered}{218} \end{array} $$Because the remainder $ \left( \color{red}{ 218 } \right) $ is not zero, we conclude that the $ x+3 $ is not a factor of $ 22x^{2}-7x-1$.
Explanation
First we need to create a synthetic division table.
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-3}&22&-7&-1\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-3&\color{orangered}{ 22 }&-7&-1\\& & & \\ \hline &\color{orangered}{22}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 22 } = \color{blue}{ -66 } $.
$$ \begin{array}{c|rrr}\color{blue}{-3}&22&-7&-1\\& & \color{blue}{-66} & \\ \hline &\color{blue}{22}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \left( -66 \right) } = \color{orangered}{ -73 } $
$$ \begin{array}{c|rrr}-3&22&\color{orangered}{ -7 }&-1\\& & \color{orangered}{-66} & \\ \hline &22&\color{orangered}{-73}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -73 \right) } = \color{blue}{ 219 } $.
$$ \begin{array}{c|rrr}\color{blue}{-3}&22&-7&-1\\& & -66& \color{blue}{219} \\ \hline &22&\color{blue}{-73}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 219 } = \color{orangered}{ 218 } $
$$ \begin{array}{c|rrr}-3&22&-7&\color{orangered}{ -1 }\\& & -66& \color{orangered}{219} \\ \hline &\color{blue}{22}&\color{blue}{-73}&\color{orangered}{218} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 218 }\right)$.