A simpler way to solve this determinant equation-$2$ 
Question Statement:-
  Solve the following determinant equation 
  $$\begin{vmatrix}
x & 2 & 3 \\
6 & x+4 & 4 \\
7 & 8 & x+8 \\
\end{vmatrix}=0$$


My Solution:-
$$\begin{vmatrix}
x & 2 & 3 \\
6 & x+4 & 4 \\
7 & 8 & x+8 \\
\end{vmatrix}=
\begin{vmatrix}
x+13 & x+14 & x+15 \\
6 & x+4 & 4 \\
7 & 8 & x+8 \\
\end{vmatrix}\tag{$R_1\rightarrow R_1+R_2+R_3$}$$
$$(x+13)\begin{vmatrix}
1 & 1 & 1 \\
6 & x+4 & 4 \\
7 & 8 & x+8 \\
\end{vmatrix}+
\begin{vmatrix}
0 & 1 & 2 \\
6 & x+4 & 4 \\
7 & 8 & x+8 \\
\end{vmatrix}\tag{1}$$
Now lets solve the two matrices obtained in the last step above seperately.
$$(x+13)\begin{vmatrix}
1 & 1 & 1 \\
6 & x+4 & 4 \\
7 & 8 & x+8 \\
\end{vmatrix}$$
$$=(x+13)\begin{vmatrix}
0 & 0 & 1 \\
2 & x & 4 \\
-(x+1) & -x & x+8 \\
\end{vmatrix}\hspace{3cm}(C_1\rightarrow C_1-C_3, C_2\rightarrow C_2-C_3)$$
$$=x(x+13)\begin{vmatrix}
0 & 0 & 1 \\
2 & 1 & 4 \\
-(x+1) & -1 & x+8 \\
\end{vmatrix}=x(x+13)(x-1)$$
And the second matrix can be simplified as 
$$\begin{vmatrix}
0 & 1 & 0 \\
6 & x+4 & -4-2x \\
7 & 8 & x-8 \\
\end{vmatrix}\tag{$C_3\rightarrow C_3-C_2$}$$
$$=-1\times(6x-48+28+14x)=-20(x-1)$$
So, we get 
$$\begin{vmatrix}
x & 2 & 3 \\
6 & x+4 & 4 \\
7 & 8 & x+8 \\
\end{vmatrix}=0\implies x(x+13)(x-1)-20(x-1)=0\\
\implies (x-1)(x^2+13x-20)=0\implies x=1,\dfrac{-13\pm\sqrt{249}}{2}$$
As can be seen, my solution is way long and it is no better than opening the determinant. So if anyone can provide me with a better, simpler and a more intuitive solution I will be very thankful. 

Your solution is fine but here is a more simple way to solve this problem:
$$\begin{vmatrix}
x & 2 & 3 \\
6 & x+4 & 4 \\
7 & 8 & x+8 \\
\end{vmatrix} $$
= $$-\begin{vmatrix}-\begin{pmatrix}
0 & 2 & 3 \\
6 & 4 & 4 \\
7 & 8 & 8 \\
\end{pmatrix} - x I)\end{vmatrix} = 0$$ where $I$ is the identity matrix.
It lookes similar to the equation for eigenvalue.
So $x$ is eigenvalue for matrix:$$A :=\begin{pmatrix}
0 & -2 & -3 \\
-6 & -4 & -4 \\
-7 & -8 & -8 \\
\end{pmatrix}$$
Because $trace(A) = -12$ and $det(A) = -20$ so the characteristic polynomial of form: $-x^3 -12 x^2 +B x -20$. Where $B$ is an real number.
How to find $B$?
Let $x = 2$ into $det(-A-xI)$,then $det(-A-xI) = -10 = -76+2B \implies B=33$.
So it becomes solve this polynomial:$-x^3 -12 x^2 +33 x -20=0$
EDIT: Another way to find B is to use $x = 1$ as subsitution, as Francesco suggested in comment. Since the row become dependent then the determinant of matrix becomes zero so $-1-12-20+B =0 \implies B = 33$ 
