The given equation is:

$2^{2x} + 3^{2x} = 4$

Let's solve this step by step.

Step 1: Substitute $y = 2^x$

We start by substituting $y = 2^x$. Therefore, $2^{2x} = (2^x)^2 = y^2$. Now, let's handle $3^{2x}$.

Step 2: Express $3^{2x}$ in terms of $y$

Now, $3^{2x}$ is a bit more complicated. We can leave it as it is but remember that it's independent of $y$. The equation becomes:

$y^2 + 3^{2x} = 4$

Step 3: Solve using possible scenarios

We try specific values of $x$:

Scenario 1: $x = 0$

$2^{2(0)} + 3^{2(0)} = 1 + 1 = 2$ Does not work.

Scenario 2: $x = 1$

Then check if $x=2$:

Generalize by seeking Solution which may include logs: