For finding the solution to something like this, it makes sense to write a function of x that is zero at the solution value. That function might be obtained by subtracting the right side from the equation.
f(x) = (2^-x +1) - (5^x +2)
Now, we want to find x so that f(x) = 0.
For the offered choices, we get
f(-1.50) ≈ 1.74
f(-0.50) ≈ -0.03
f(0.75) ≈ -3.75
f(2.50) ≈ -56.72
The obvious choice is
x = -0.50
