If x = 1 + 2 + 3 and y = 1 + 2 - 3 , then the value of x^2 + 4xy + y^2x + y is?

Answer: Option D

$$\begin{array}{rl}& \text{According to the question,}\\ & x=1+\sqrt{2}+\sqrt{3}.....(i)\\ & y=1+\sqrt{2}-\sqrt{3}.....(ii)\\ & \Rightarrow \frac{x^2+4xy+y^2}{x+y}\\ & \Rightarrow \frac{{\left(x+y\right)}^2+2xy}{x+y}\\ & \text{From equation (i)}+\text{(ii)}\\ & \Rightarrow x+y=2+2\sqrt{2}\\ & xy={\left(1+\sqrt{2}\right)}^2-{\left(\sqrt{3}\right)}^2\\ & \Rightarrow xy=3+2\sqrt{2}-3\\ & \Rightarrow xy=2\sqrt{2}\\ & \text{So, }\frac{{\left(x+y\right)}^2+2xy}{x+y}\\ & =\frac{{\left(2+2\sqrt{2}\right)}^2+2\times 2\sqrt{2}}{2+2\sqrt{2}}\\ & =\frac{4+8+8\sqrt{2}+4\sqrt{2}}{2+2\sqrt{2}}\\ & =\frac{12+12\sqrt{2}}{2+2\sqrt{2}}\\ & =\frac{12\left(1+\sqrt{2}\right)}{2\left(1+\sqrt{2}\right)}\\ & =6\end{array}$$