Expanding the left side:

(x-4)^2 - 6 = (x-4)(x-4) - 6
= x^2 - 4x - 4x + 16 - 6
= x^2 - 8x + 10

Expanding the right side:

(2+x)^2 = (2+x)(2+x)
= 2*2 + 2x + 2x + x^2
= 4 + 4x + x^2

Now we need to equate the two sides:

x^2 - 8x + 10 = 4 + 4x + x^2

Now, let's solve this equation:

x^2 - 8x + 10 = 4 + 4x + x^2
=> -8x + 10 = 4 + 4x
=> -8x - 4x = 4 - 10
=> -12x = -6
=> x = 1/2

Therefore, the solution to the equation is x = 1/2.