To solve the equation 2^x + 1 + x^2 = 32, we need to find the value of x that satisfies the equation.

First, let's subtract 1 from both sides of the equation:

2^x + x^2 = 31

Next, let's subtract 31 from both sides of the equation:

2^x + x^2 - 31 = 0

Now, we have a quadratic equation. To solve it, we can try factoring or using the quadratic formula. However, in this case, factoring is not straightforward, so let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 1, b = 0, and c = -31. Plugging these values into the quadratic formula, we get:

x = (0 ± √(0^2 - 4(1)(-31))) / (2(1))

Simplifying further:

x = ± √(0 - (-124)) / 2

x = ± √(124) / 2

x = ± √(31)

Therefore, the solutions to the equation 2^x + 1 + x^2 = 32 are x = √(31) and x = -√(31).