To solve this equation, let's first simplify the equation by expanding the terms:

x^2 + 1/x^2 + 2(x + 1/x) = 6
x^2 + 1/x^2 + 2x + 2/x = 6

Now, let's combine the terms that involve x and 1/x:

x^2 + 1/x^2 + 2x + 2/x = 6
(x^2 + 2x) + (1/x^2 + 2/x) = 6
x(x + 2) + 1/x(x + 2) = 6
(x + 1/x)(x + 2) = 6

Now, we have a quadratic equation in the form of (x + 1/x)(x + 2) = 6. Let's solve for x by expanding the left side of the equation:

(x + 1/x)(x + 2) = 6
x^2 + 2x + 1 = 6
x^2 + 2x - 5 = 0

Now, we have a quadratic equation x^2 + 2x - 5 = 0. We can solve this quadratic equation using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values of a = 1, b = 2, and c = -5 into the formula, we get:

x = (-2 ± √(2^2 - 4(1)(-5))) / (2(1))
x = (-2 ± √(4 + 20)) / 2
x = (-2 ± √24) / 2
x = (-2 ± 2√6) / 2
x = -1 ± √6

Therefore, the solutions to the equation x^2 + 1/x^2 + 2(x + 1/x) = 6 are x = -1 + √6 and x = -1 - √6.