To solve for x, we can rewrite the equation as sin 2x + (cos x + sin x) + 1 = 0.

Now, we can rewrite sin 2x in terms of sin x and cos x using the double angle identity sin 2x = 2sin x cos x.

So, our equation becomes 2sin x cos x + (cos x + sin x) + 1 = 0.

Now, let's rearrange the terms: 2sin x cos x + cos x + sin x + 1 = 0.

Factor out sin x and cos x: sin x(2cos x + 1) + cos x(1 + 1) = 0.

Now, we have: sin x(2cos x + 1) + 2cos x = 0.

Divide through by 2: sin x(2cos x + 1)/2 + cos x = 0.

Now, we can simplify and factor out a sin x: sin x(cos x + 1) + cos x = 0.

Now, we can further simplify: sin x cos x + sin x + cos x = 0.

Re-arranging the terms: sin x cos x + sin x = -cos x.

Divide by sin x: cos x + 1 = -cot x.

Substitute cos x = -1 - sin x into the equation: -1 - sin x + 1 = -cot x.

Therefore, the solution is x = n * π, where n is any integer.