To solve the equation \( 7^{2x} + 7^{2x+3} = 344 \), we can first simplify the equation.

We can rewrite the term \( 7^{2x+3} \):

\[
7^{2x+3} = 7^{2x} \cdot 7^3 = 7^{2x} \cdot 343
\]

Substituting this back into the original equation gives:

\[
7^{2x} + 343 \cdot 7^{2x} = 344
\]

This can be factored since \( 7^{2x} \) is common:

\[
7^{2x} (1 + 343) = 344
\]

This simplifies to:

\[
7^{2x} \cdot 344 = 344
\]

Now we divide both sides by 344 (assuming \( 344 \neq 0 \)):

\[
7^{2x} = 1
\]

The equation \( 7^{2x} = 1 \) implies that:

\[
2x = 0
\]

So,

\[
x = 0
\]

Thus, the solution to the equation is:

\[
\boxed{0}
\]