Step-by-Step Solution:
1. Isolate the square root and square both sides:

$$
\(\begin{gathered}
\sqrt{8 x^2+17}=3 x-2 \\
\left(\sqrt{8 x^2+17}\right)^2=(3 x-2)^2
\end{gathered}\)
$$

2. Expand the equation:

$$
\(8 x^2+17=9 x^2-12 x+4\)
$$

3. Move all terms to one side to form a quadratic equation:

Subtract $\(8 x^2\)$ and 17 from both sides:

$$
\(0=x^2-12 x-13\)
$$

4. Factor the quadratic equation:

We look for two numbers that multiply to -13 and add to -12 :

$$
\(0=(x-13)(x+1)\)
$$


So, the potential solutions are:

$$
\(x=13 \quad \text { or } \quad x=-1\)
$$

5. Check for extraneous solutions:

Because we squared both sides, we must verify the solutions in the original equation (specifically, the right side $3 x-2$ must be non-negative).
- Check $\(x=13\)$ :

$$
\(\begin{aligned}
& \text { LHS: } \sqrt{8(13)^2+17}=\sqrt{8(169)+17}=\sqrt{1352+17}=\sqrt{1369}=37 \\
& \text { RHS: } 3(13)-2=39-2=37 \\
& 37=37 \text { (Valid) }
\end{aligned}\)
$$

- Check $\(x=-1\)$ :

$$
\(\begin{aligned}
& \text { LHS: } \sqrt{8(-1)^2+17}=\sqrt{8+17}=\sqrt{25}=5 \\
& \text { RHS: } 3(-1)-2=-3-2=-5 \\
& 5 \neq-5 \text { (Extraneous) }
\end{aligned}\)
$$

6. Final Calculation:

Since $\(x=13\)$ :

$$
\(2 x=2 \times 13=26\)
$$