How do you use the synthetic division method to divide $\left( 5{{x}^{3}}+18{{x}^{2}}+7x-6 \right)\div \left( x+3 \right)$?

Hint: We start solving the problem by finding the divisor and dividend in the given division. We then write the coefficients of the dividend by following the properties of the synthetic division. We then write the divisors and quotients by following the steps of the synthetic division. We then complete the synthetic division and then find the remainder and quotient of the given division process to get the required answer.

Complete step by step solution:
According to the problem, we are asked to divide the given division $\left( 5{{x}^{3}}+18{{x}^{2}}+7x-6 \right)\div \left( x+3 \right)$ using the synthetic division method.
We have the divisor $\left( x+3 \right)$ and the dividend as $\left( 5{{x}^{3}}+18{{x}^{2}}+7x-6 \right)$.
We first write the coefficients of the given dividend as shown below:
$\left| \!{\underline {\,
  \begin{matrix}
   \begin{matrix}
   5 & 18 & 7 & -6 \\
\end{matrix} \\
   {} \\
\end{matrix} \,}} \right. $.
Now, let us equate the divisor $\left( x+3 \right)$ to 0 which gives $x=-3$.
$-3\left| \!{\underline {\,
  \begin{matrix}
   \begin{matrix}
   5 & 18 & 7 & -6 \\
\end{matrix} \\
   {} \\
\end{matrix} \,}} \right. $.
Now, the first coefficient of the dividend is dropped down as shown below:
$\begin{align}
  & -3\left| \!{\underline {\,
  \begin{matrix}
   5 & 18 & 7 & -6 \\
   0 & {} & {} & {} \\
\end{matrix} \,}} \right. \\
 & \text{ }\begin{matrix}
   5 & {} & {} & {} \\
\end{matrix} \\
\end{align}$.
We perform the remaining division as shown below:
$\begin{align}
  & -3\left| \!{\underline {\,
  \begin{matrix}
   5 & 18 & 7 & -6 \\
   0 & -15 & -9 & 6 \\
\end{matrix} \,}} \right. \\
 & \text{ 5 3 }-2\text{ 0} \\
\end{align}$.
We know that the last term in the synthetic division is the remainder of the division. So, the remainder is 0.
Now, the quotient is $5{{x}^{2}}+3x-2$.
So, we have found the quotient and remainder of the division $\left( 5{{x}^{3}}+18{{x}^{2}}+7x-6 \right)\div \left( x+3 \right)$ as $5{{x}^{2}}+3x-2$ and 0.
$\therefore $ The quotient and remainder of the division $\left( 5{{x}^{3}}+18{{x}^{2}}+7x-6 \right)\div \left( x+3 \right)$ is $5{{x}^{2}}+3x-2$ and 0.

Note: We should perform each step carefully in order to avoid confusion and calculation mistakes while solving this problem. We can also perform the given division by making use of the long division method which will also give similar results. Similarly, we can expect the problems to factorize the obtained quotients from the division just performed.