1. **Problem Statement:** Simplify the expression $$\left(4 \sqrt{5} + 3 \right)^2$$. 2. **Formula Used:** The square of a binomial $$ (a + b)^2 = a^2 + 2ab + b^2 $$. 3. **Identify terms:** Here, $$a = 4 \sqrt{5}$$ and $$b = 3$$. 4. **Calculate each term:** - $$a^2 = (4 \sqrt{5})^2 = 4^2 \times (\sqrt{5})^2 = 16 \times 5 = 80$$ - $$2ab = 2 \times 4 \sqrt{5} \times 3 = 24 \sqrt{5}$$ - $$b^2 = 3^2 = 9$$ 5. **Combine all terms:** $$80 + 24 \sqrt{5} + 9 = 89 + 24 \sqrt{5}$$ 6. **Final answer:** $$\left(4 \sqrt{5} + 3 \right)^2 = 89 + 24 \sqrt{5}$$ This expression is simplified and cannot be reduced further because $$\sqrt{5}$$ is irrational.