1. **State the problem:** Simplify the expression $$\frac{1+2\cos x \sin x}{\cos x \sin x + \cos^2 x}$$. 2. **Rewrite the numerator:** The numerator is $$1 + 2\cos x \sin x$$. 3. **Rewrite the denominator:** The denominator is $$\cos x \sin x + \cos^2 x$$. 4. **Use trigonometric identities:** Recall that $$\sin(2x) = 2\sin x \cos x$$, so $$2\cos x \sin x = \sin(2x)$$. 5. **Rewrite numerator using identity:** Numerator becomes $$1 + \sin(2x)$$. 6. **Factor denominator:** Factor out $$\cos x$$ from the denominator: $$\cos x \sin x + \cos^2 x = \cos x (\sin x + \cos x)$$. 7. **Rewrite the expression:** $$\frac{1 + \sin(2x)}{\cos x (\sin x + \cos x)}$$. 8. **Final simplified form:** The expression is simplified to $$\frac{1 + \sin(2x)}{\cos x (\sin x + \cos x)}$$. This is the simplified form unless further context or constraints are given.