To solve this equation, we first need to expand the right side of the equation:

(x+6)^2 = x^2 + 12x + 36

Now we can substitute this into our original equation:

2x^2 + 11x + 34 = x^2 + 12x + 36

Next, we want to set the equation equal to zero by moving all terms to one side:

2x^2 + 11x + 34 - x^2 - 12x - 36 = 0

Simplify:

x^2 - x - 2 = 0

Now we have a quadratic equation that we can solve using factoring or the quadratic formula. Factoring this equation, we get:

(x - 2)(x + 1) = 0

Setting each factor to zero gives us the solutions:

x - 2 = 0 or x + 1 = 0

x = 2 or x = -1

Therefore, the solutions to the equation 2x^2 + 11x + 34 = (x+6)^2 are x = 2 and x = -1.