To solve this equation, we can make a substitution by letting (Y = (X-4)^2). Then the equation becomes:

(Y^2 + 2Y = 8)

Now we have a quadratic equation in terms of Y. We can rearrange it to get:

(Y^2 + 2Y - 8 = 0)

Now we can factor this quadratic equation to get:

((Y + 4)(Y - 2) = 0)

Setting each factor to zero gives:

(Y + 4 = 0) or (Y - 2 = 0)

(Y = -4) or (Y = 2)

Now we substitute back our original variable equation to solve for X:

Case 1: If (Y = -4), then ((X-4)^2 = -4)

(X - 4 = \pm 2i)

(X = 4 \pm 2i)

Case 2: If (Y = 2), then ((X-4)^2 = 2)

(X - 4 = \pm \sqrt{2})

(X = 4 \pm \sqrt{2})

Therefore, the solutions to the equation are:

[X = 4 \pm 2i \text{ or } X = 4 \pm \sqrt{2}]