1. **State the problem:** We need to find the value of $$(a+b)^2$$ given that $$a+b=3$$ and $$ab=-2$$. 2. **Recall the identity:** The square of a binomial sum is given by $$ (a+b)^2 = a^2 + 2ab + b^2 $$ 3. **Rewrite using the given sum:** We can express $$a^2 + b^2$$ in terms of $$a+b$$ and $$ab$$ using the identity $$ a^2 + b^2 = (a+b)^2 - 2ab $$ 4. **Calculate using the given values:** Since $$a+b=3$$, we have $$ (a+b)^2 = 3^2 = 9 $$ Given $$ab = -2$$, then $$ a^2 + b^2 = 9 - 2(-2) = 9 + 4 = 13 $$ 5. **Find the value of $$(a+b)^2$$:** By substituting back, $$ (a+b)^2 = a^2 + 2ab + b^2 = (a^2 + b^2) + 2ab = 13 + 2(-2) = 13 - 4 = 9 $$ 6. **Check:** Since we had $$a+b=3$$, the square of the sum is directly $$3^2=9$$, which matches our calculation. **Final answer:** $$(a+b)^2 = 9$$