Divide using synthetic division $$ ( 6x^{5}-5x^{3}-7x+5 ) \div ( x-2 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}2&6&0&-5&0&-7&5\\& & 12& 24& 38& 76& \color{black}{138} \\ \hline &\color{blue}{6}&\color{blue}{12}&\color{blue}{19}&\color{blue}{38}&\color{blue}{69}&\color{orangered}{143} \end{array} $$The solution is:
$$ \dfrac{ 6x^{5}-5x^{3}-7x+5 }{ x-2 } = \color{blue}{6x^{4}+12x^{3}+19x^{2}+38x+69} ~+~ \dfrac{ \color{red}{ 143 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&6&0&-5&0&-7&5\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}2&\color{orangered}{ 6 }&0&-5&0&-7&5\\& & & & & & \\ \hline &\color{orangered}{6}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 6 } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&6&0&-5&0&-7&5\\& & \color{blue}{12} & & & & \\ \hline &\color{blue}{6}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 12 } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrrrrr}2&6&\color{orangered}{ 0 }&-5&0&-7&5\\& & \color{orangered}{12} & & & & \\ \hline &6&\color{orangered}{12}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 12 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&6&0&-5&0&-7&5\\& & 12& \color{blue}{24} & & & \\ \hline &6&\color{blue}{12}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 24 } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrrrrr}2&6&0&\color{orangered}{ -5 }&0&-7&5\\& & 12& \color{orangered}{24} & & & \\ \hline &6&12&\color{orangered}{19}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 19 } = \color{blue}{ 38 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&6&0&-5&0&-7&5\\& & 12& 24& \color{blue}{38} & & \\ \hline &6&12&\color{blue}{19}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 38 } = \color{orangered}{ 38 } $
$$ \begin{array}{c|rrrrrr}2&6&0&-5&\color{orangered}{ 0 }&-7&5\\& & 12& 24& \color{orangered}{38} & & \\ \hline &6&12&19&\color{orangered}{38}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 38 } = \color{blue}{ 76 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&6&0&-5&0&-7&5\\& & 12& 24& 38& \color{blue}{76} & \\ \hline &6&12&19&\color{blue}{38}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 76 } = \color{orangered}{ 69 } $
$$ \begin{array}{c|rrrrrr}2&6&0&-5&0&\color{orangered}{ -7 }&5\\& & 12& 24& 38& \color{orangered}{76} & \\ \hline &6&12&19&38&\color{orangered}{69}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 69 } = \color{blue}{ 138 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&6&0&-5&0&-7&5\\& & 12& 24& 38& 76& \color{blue}{138} \\ \hline &6&12&19&38&\color{blue}{69}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 138 } = \color{orangered}{ 143 } $
$$ \begin{array}{c|rrrrrr}2&6&0&-5&0&-7&\color{orangered}{ 5 }\\& & 12& 24& 38& 76& \color{orangered}{138} \\ \hline &\color{blue}{6}&\color{blue}{12}&\color{blue}{19}&\color{blue}{38}&\color{blue}{69}&\color{orangered}{143} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{4}+12x^{3}+19x^{2}+38x+69 } $ with a remainder of $ \color{red}{ 143 } $.