1. **State the problem:** Simplify the expression $2 - 2\sin A - 2\sin A \cos A + 2\cos A$. 2. **Group like terms:** Group terms involving $\sin A$ and $\cos A$ separately: $$2 - 2\sin A - 2\sin A \cos A + 2\cos A = 2 + 2\cos A - 2\sin A - 2\sin A \cos A$$ 3. **Factor common terms:** Factor $2$ out: $$= 2(1 + \cos A - \sin A - \sin A \cos A)$$ 4. **Factor by grouping:** Group terms inside parentheses: $$= 2[(1 + \cos A) - \sin A(1 + \cos A)]$$ 5. **Factor out $(1 + \cos A)$:** $$= 2(1 + \cos A)(1 - \sin A)$$ 6. **Final simplified form:** $$2(1 + \cos A)(1 - \sin A)$$ This shows the expression factored into a product of simpler terms, which can be useful for further evaluation or solving.