1. The problem asks us to find the value of $$\left(4\sqrt{4} + 2\sqrt{3} - \sqrt{49 + 8 \sqrt{3}} \right)^2$$. 2. Start by simplifying the terms inside the parentheses: $$4\sqrt{4} = 4 \times 2 = 8$$ So the expression becomes: $$\left(8 + 2\sqrt{3} - \sqrt{49 + 8 \sqrt{3}} \right)^2$$ 3. Next, simplify the nested radical $$\sqrt{49 + 8 \sqrt{3}}$$. We try to express it as $$\sqrt{(a + b)^2} = a + b$$ where $$a$$ and $$b$$ involve square roots. Assume: $$\sqrt{49 + 8 \sqrt{3}} = \sqrt{x} + \sqrt{y}$$ Then, $$49 + 8 \sqrt{3} = (\sqrt{x} + \sqrt{y})^2 = x + y + 2\sqrt{xy}$$ Matching components: $$x + y = 49$$ $$2\sqrt{xy} = 8 \sqrt{3} \implies \sqrt{xy} = 4 \sqrt{3}$$ Square both sides: $$xy = 16 \times 3 = 48$$ 4. Solve the system: $$x + y = 49$$ $$xy = 48$$ This suggests $x$ and $y$ are roots of the quadratic equation: $$t^2 - 49t + 48 = 0$$ The discriminant $$\Delta = 49^2 - 4 \times 48 = 2401 - 192 = 2209$$ $$\sqrt{2209} = 47$$ Roots are: $$t = \frac{49 \pm 47}{2}$$ which gives $$t_1 = \frac{49 + 47}{2} = 48$$ $$t_2 = \frac{49 - 47}{2} = 1$$ 5. So, $$x=48$$ and $$y=1$$. Thus: $$\sqrt{49 + 8 \sqrt{3}} = \sqrt{48} + \sqrt{1} = 4 \sqrt{3} + 1$$ 6. Substitute back: $$\left(8 + 2 \sqrt{3} - (4 \sqrt{3} + 1) \right)^2 = \left(8 + 2 \sqrt{3} - 4 \sqrt{3} - 1\right)^2 = \left(7 - 2 \sqrt{3} \right)^2$$ 7. Expand $$\left(7 - 2 \sqrt{3} \right)^2$$: $$= 7^2 - 2 \times 7 \times 2 \sqrt{3} + (2 \sqrt{3})^2 = 49 - 28 \sqrt{3} + 4 \times 3 = 49 - 28 \sqrt{3} + 12 = 61 - 28 \sqrt{3}$$ 8. This expression does not match any option exactly as is. Let's check the original choices. Since the problem is multiple choice, let's test if $$61 - 28 \sqrt{3}$$ equals any listed options. Calculate approximate value of $$61 - 28 \sqrt{3}$$: $$\sqrt{3} \approx 1.732$$ $$28 \times 1.732 = 48.496$$ $$61 - 48.496 = 12.504$$ None of the options equal 12.504 directly. 9. Review the problem to check if the expression might correspond or simplify differently. Notice that the problem asks for the value of the squared expression. One option presented with format $$4 \sqrt{3} + 3$$ is close to our intermediate result. 10. The final result after calculation is $$61 - 28 \sqrt{3}$$. Hence, the value of the expression $$\left(4\sqrt{4} + 2\sqrt{3} - \sqrt{49 + 8 \sqrt{3}} \right)^2$$ is $$61 - 28 \sqrt{3}$$. This does not exactly correspond to the given options listed, so none of the multiple-choice answers match the computed exact value. Thus, the exact simplified answer is $$61 - 28 \sqrt{3}$$.