1. **State the problem:** Find the value of $$\left(4\sqrt{4} + 2\sqrt{3} - \sqrt{49 + 8\sqrt{3}}\right)^2.$$\n\n2. **Simplify the terms inside the parentheses:**\n- Calculate $$4\sqrt{4} = 4 \times 2 = 8.$$\n- The term $$2\sqrt{3}$$ remains as is.\n- Next, simplify $$\sqrt{49 + 8\sqrt{3}}$$ if possible.\n\n3. **Simplify $$\sqrt{49 + 8\sqrt{3}}$$:**\nAssume $$\sqrt{49 + 8\sqrt{3}} = \sqrt{a} + \sqrt{b}$$ where $$a$$ and $$b$$ are positive numbers.\n\nThen, $$(\sqrt{a} + \sqrt{b})^2 = a + b + 2\sqrt{ab} = 49 + 8\sqrt{3}.$$\nHence, we get the system:\n$$a + b = 49,$$\n$$2\sqrt{ab} = 8\sqrt{3}.$$\nFrom the second, $$2\sqrt{ab} = 8\sqrt{3} \implies \sqrt{ab} = 4\sqrt{3} \implies ab = 16 \times 3 = 48.$$\n\n4. **Solve for $$a$$ and $$b$$:**\nFrom $a+b=49$ and $ab=48$, solve the quadratic:\n$$x^2 - 49x + 48 = 0.$$\nCalculate the discriminant:\n$$\Delta = 49^2 - 4 \times 48 = 2401 - 192 = 2209.$$\nCheck if $$\sqrt{2209} = 47.$$\nTherefore, solutions are:\n$$x = \frac{49 \pm 47}{2}.$$\n\nPossible values:\n- $$x_1 = \frac{49+47}{2} = \frac{96}{2} = 48,$$\n- $$x_2 = \frac{49-47}{2} = \frac{2}{2} = 1.$$\n\nSo, $$a = 48$$ and $$b = 1$$ (or vice versa).\n\n5. **Substitute back:**\n$$\sqrt{49 + 8\sqrt{3}} = \sqrt{48} + \sqrt{1} = 4\sqrt{3} + 1.$$\n\n6. **Rewrite the original expression:**\n$$\left(8 + 2\sqrt{3} - (4\sqrt{3} + 1)\right)^2 = \left(8 + 2\sqrt{3} - 4\sqrt{3} - 1\right)^2 = \left(7 - 2\sqrt{3}\right)^2.$$\n\n7. **Expand $$\left(7 - 2\sqrt{3}\right)^2$$:**\n$$ = 7^2 - 2 \times 7 \times 2\sqrt{3} + (2\sqrt{3})^2 = 49 - 28\sqrt{3} + 4 \times 3 = 49 - 28\sqrt{3} + 12 = 61 - 28\sqrt{3}.$$\n\n8. **Result:**\nThe value of the expression is $$61 - 28\sqrt{3}$$ which does not directly match any given choices. However, to verify options, check approximate value:\n$$28\sqrt{3} \approx 28 \times 1.732 = 48.5,$$\nso $$61 - 48.5 = 12.5.$$\nNo option matches 12.5 exactly. Let's try another approach since the options include $$4\sqrt{3}+3$$ which is close to some intermediate values.\n\n\n**Recheck step 5:** Simplify $$\sqrt{49+8\sqrt{3}}$$ as $$m + n\sqrt{3}$$ for real numbers $$m$$ and $$n$$.\nThen, $$(m + n\sqrt{3})^2 = m^2 + 2mn\sqrt{3} + 3n^2 = 49 + 8\sqrt{3}.$$\nEquate rational and irrational parts:\n$$m^2 + 3n^2 = 49,$$\n$$2mn = 8 \implies mn=4.$$\nTry integer pairs $(m,n)$ with $mn=4$: (1,4), (2,2), (4,1).\nTest (7,1), not possible since $$7 \times 1 = 7\ne4.$$\nTry (4,1):\n$$m^2 + 3n^2 = 4^2 + 3 \times 1^2 = 16+3=19 e 49.$$\nTry (2,2):\n$$2^2 + 3 \times 2^2 = 4 + 3 \times 4 = 4 + 12 = 16 e 49.$$\nTry (1,4):\n$$1 + 3 \times 16 = 1 + 48 = 49,$$\nand $$2mn = 2 \times 1 \times 4 = 8.$$ Perfect match.\n\nThus, $$\sqrt{49 + 8\sqrt{3}} = 1 + 4\sqrt{3}.$$\n\n9. **Substitute corrected value:**\n\nOriginal expression:\n$$\left(8 + 2\sqrt{3} - (1 + 4\sqrt{3})\right)^2 = (8 - 1 + 2\sqrt{3} - 4\sqrt{3})^2 = (7 - 2\sqrt{3})^2.$$\n\n10. **Expand again:**\n(As before) $$= 49 - 28\sqrt{3} + 12 = 61 - 28\sqrt{3}.$$\n\nNo direct match appears in options, let's approximate again: $$61 - 28\sqrt{3} \approx 61 - 48.5 = 12.5.$$\n\n**Check options:**\n- $$3\sqrt{3} \approx 5.2,$$\n- $$9,$$\n- $$6,$$\n- $$4\sqrt{3} \approx 6.93,$$\n- $$4\sqrt{3} + 3 \approx 6.93 + 3 = 9.93.$$\n\nClosest to 12.5 is none, meaning the expression does not match any options exactly.\n\n**Possibility:** The question asks for the value of the square, which we calculated as $$61 - 28\sqrt{3}$$. The given options may refer to the inside of the parentheses or a simplification error in the problem statement.\n\nSince none match, the simplified form $$\boxed{61 - 28\sqrt{3}}$$ is the exact value.