To solve this equation, we can use the double angle identity for sine, which states that:

sin(2θ) = 2sinθcosθ

Given the equation:

2sin^2(7π/2 + x) = 5sinx + 4

We can rewrite sin^2(7π/2 + x) as:

sin^2(7π/2 + x) = 1 - cos^2(7π/2 + x)

Now, let's plug this into the equation:

2(1 - cos^2(7π/2 + x)) = 5sinx + 4
2 - 2cos^2(7π/2 + x) = 5sinx + 4
-2cos^2(7π/2 + x) = 5sinx + 2
cos^2(7π/2 + x) = -(5sinx + 2)/2

Now, we can use the double angle identity for sine to convert cos^2(7π/2 + x) into a sine function:

sin^2(7π/2 + x) = 1 - cos^2(7π/2 + x)
sin^2(7π/2 + x) = 1 - (-(5sinx + 2)/2)
sin^2(7π/2 + x) = 1 + (5sinx + 2)/2

Now, we have sin^2(7π/2 + x) in terms of sinx. We can further simplify this expression and solve for sinx.