To solve the equation 2^x + 1 . 2^2x - 24 = 100, we can simplify it step by step.

First, let's simplify the expression 2^2x. Since 2^2x is equal to (2^x)^2, we can rewrite the equation as:

2^x + 1 . (2^x)^2 - 24 = 100

Next, let's simplify the expression 1 . (2^x)^2. Multiplying 1 by any number does not change the value, so we can simplify it to:

(2^x)^2 - 24 = 100

Now, let's simplify the expression (2^x)^2. Since (2^x)^2 is equal to 2^(2x), we can rewrite the equation as:

2^(2x) - 24 = 100

Next, let's add 24 to both sides of the equation to isolate the term with the exponent:

2^(2x) = 100 + 24

2^(2x) = 124

Now, let's take the logarithm of both sides of the equation to solve for x. We can use any base for the logarithm, but let's use base 2 since we have a 2 in the equation:

log2(2^(2x)) = log2(124)

2x = log2(124)

Finally, divide both sides of the equation by 2 to solve for x:

x = log2(124) / 2

Using a calculator, we can find that log2(124) is approximately 6.9542. Therefore, x is approximately:

x ≈ 6.9542 / 2

x ≈ 3.4771