Let's simplify the equation step by step:

(2^x + 1)^2 + 3(2^2 + 1) = 2^2 * 10

First, simplify the expression inside the parentheses:

(2^x + 1)^2 + 3(4 + 1) = 4 * 10

Next, simplify the exponent:

(2^x + 1)(2^x + 1) + 3(5) = 4 * 10

Expand the squared term:

(2^x)^2 + 2(2^x)(1) + 1 + 15 = 40

Simplify further:

2^(2x) + 2^(x+1) + 16 = 40

Now, let's isolate the term with the exponent:

2^(2x) + 2^(x+1) = 40 - 16

2^(2x) + 2^(x+1) = 24

Next, let's simplify the exponent in the second term:

2^(2x) + 2^x * 2^1 = 24

2^(2x) + 2^(x+1) = 24

Now, let's combine the terms with the same base:

2^(2x) + 2 * 2^x = 24

Finally, let's factor out the common factor of 2^x:

2^x(2^x + 2) = 24

Now, we can solve for 2^x:

2^x = 24 / (2^x + 2)

This is as far as we can simplify the equation without knowing the value of x.