1. **State the problem:** Given that $m+n=3$, find the value of the expression $$2m^2 + 4mn + 2n^2 - 6.$$ 2. **Rewrite and simplify the expression:** Notice we can group the terms as $$2(m^2 + 2mn + n^2) - 6.$$ 3. **Recognize the perfect square:** The expression inside the parentheses is a perfect square: $$m^2 + 2mn + n^2 = (m+n)^2.$$ 4. **Substitute the given sum:** Since $m+n = 3$, then $$(m+n)^2 = 3^2 = 9.$$ 5. **Calculate the value:** Substitute into the expression: $$2 imes 9 - 6 = 18 - 6 = 12.$$ 6. **Conclusion:** The value of the given expression is **12**.