1. State the problem: We want to find the value of $\left(4\sqrt{4} + 2\sqrt{3} - \sqrt{49} + 8\sqrt{3}\right)^2$. 2. Simplify each square root: - $\sqrt{4} = 2$ - $\sqrt{3}$ stays as is - $\sqrt{49} = 7$ 3. Substitute the simplified values back: $$\left(4 \times 2 + 2\sqrt{3} - 7 + 8\sqrt{3}\right)^2$$ 4. Multiply and combine like terms: $$\left(8 + 2\sqrt{3} - 7 + 8\sqrt{3}\right)^2 = \left( (8 - 7) + (2\sqrt{3} + 8\sqrt{3}) \right)^2 = \left(1 + 10\sqrt{3}\right)^2$$ 5. Use the formula $(a+b)^2 = a^2 + 2ab + b^2$: $$1^2 + 2 \times 1 \times 10\sqrt{3} + (10\sqrt{3})^2 = 1 + 20\sqrt{3} + 100 \times 3$$ 6. Calculate $100 \times 3 = 300$ and sum all: $$1 + 20\sqrt{3} + 300 = 301 + 20\sqrt{3}$$ Final answer: $$\boxed{301 + 20\sqrt{3}}$$ Note: None of the provided options exactly match this result.