1. **State the problem:** Factorize the expression $$(2m + n)^2 - (2m + n)(n + 2)$$. 2. **Identify common factors:** Notice that both terms contain the factor $$(2m + n)$$. 3. **Factor out the common term:** $$ (2m + n)^2 - (2m + n)(n + 2) = (2m + n)\big((2m + n) - (n + 2)\big) $$ 4. **Simplify inside the parentheses:** $$ (2m + n) - (n + 2) = 2m + n - n - 2 = 2m - 2 $$ 5. **Factor the simplified expression:** $$ 2m - 2 = 2(m - 1) $$ 6. **Write the fully factorized form:** $$ (2m + n) \times 2(m - 1) = 2(2m + n)(m - 1) $$ **Final answer:** $$2(2m + n)(m - 1)$$