1. **State the problem:** We want to approximate the value of $y$ at $x=1.1$ given that $y=1.2$ at $x=1$ and the differential equation $$\frac{dy}{dx} = 3x + y^2.$$\n\n2. **Recall Runge's method:** Runge's method is a numerical technique to approximate solutions of differential equations. It improves on Euler's method by using an intermediate slope. The formula for one step from $x_n$ to $x_{n+1} = x_n + h$ is:\n$$k_1 = f(x_n,y_n)$$\n$$k_2 = f\left(x_n + \frac{h}{2}, y_n + \frac{h}{2}k_1\right)$$\n$$y_{n+1} = y_n + h k_2,$$\nwhere $f(x,y) = \frac{dy}{dx}$ and $h$ is the step size.\n\n3. **Identify given values:**\n- Initial point: $x_0 = 1$, $y_0 = 1.2$\n- Step size: $h = 1.1 - 1 = 0.1$\n- Differential function: $f(x,y) = 3x + y^2$\n\n4. **Calculate $k_1$:**\n$$k_1 = f(1,1.2) = 3(1) + (1.2)^2 = 3 + 1.44 = 4.44.$$\n\n5. **Calculate $k_2$:**\nFirst compute intermediate values:\n$$x_1 = 1 + \frac{0.1}{2} = 1.05,$$\n$$y_1 = 1.2 + \frac{0.1}{2} \times 4.44 = 1.2 + 0.05 \times 4.44 = 1.2 + 0.222 = 1.422.$$\nNow evaluate $f$ at these values:\n$$k_2 = f(1.05, 1.422) = 3(1.05) + (1.422)^2 = 3.15 + 2.022 = 5.172.$$\n\n6. **Calculate $y$ at $x=1.1$:**\n$$y(1.1) = y_0 + h k_2 = 1.2 + 0.1 \times 5.172 = 1.2 + 0.5172 = 1.7172.$$\n\n**Final answer:** The approximate value of $y$ at $x=1.1$ using Runge's method is $$\boxed{1.7172}.$$