1. **State the problem:** Simplify the expression $x^2 + 2x + 2xy + 4y$. 2. **Group like terms:** Group terms with common factors: $x^2 + 2x + 2xy + 4y = x^2 + 2x + 2xy + 4y$. 3. **Factor by grouping:** Group the first two and last two terms: $$ (x^2 + 2x) + (2xy + 4y) $$ 4. **Factor out common factors in each group:** $$ x(x + 2) + 2y(x + 2) $$ 5. **Factor out the common binomial factor:** $$ (x + 2)(x + 2y) $$ **Final answer:** $$ (x + 2)(x + 2y) $$