To solve the equation, let's first expand and simplify both sides:

(2x-1)^2 + (x+1)^2 = 7(x^2+1) - 2(x^2+1/2x)

Expanding the squares:

(4x^2 - 4x + 1) + (x^2 + 2x + 1) = 7x^2 + 7 - 2x^2 - 1/x

Combining like terms:

4x^2 - 4x + 1 + x^2 + 2x + 1 = 7x^2 + 7 - 2x^2 - 1/x

5x^2 - 2x + 2 = 5x^2 + 7 - 1/x

Subtracting 5x^2 from both sides:

-2x + 2 = 7 - 1/x

Adding 2x to both sides:

2 = 7 - 1/x + 2x

Combining like terms:

2 = 7 + 2x - 1/x

Subtracting 7 from both sides:

-5 = 2x - 1/x

Multiplying both sides by x:

-5x = 2x^2 - 1

Rearranging the equation:

2x^2 + 5x - 1 = 0

Now we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 2, b = 5, and c = -1. Plugging in these values:

x = (-5 ± √(5^2 - 4(2)(-1))) / (2(2))

Simplifying:

x = (-5 ± √(25 + 8)) / 4

x = (-5 ± √33) / 4

Therefore, the solutions to the equation are:

x = (-5 + √33) / 4

x = (-5 - √33) / 4