To solve this problem, we need to determine how far above the x-axis the lamp is positioned, given that it casts a shadow represented by the ellipse $x^2 + 4y^2 = 5$.

Step-by-Step Solution

1. Ellipse Analysis: The equation of the shadow's elliptical region is: $x^2 + 4y^2 = 5$ This equation can be rewritten in standard form: $\frac{x^2}{5} + \frac{y^2}{\frac{5}{4}} = 1$ This represents an ellipse centered at the origin, with semi-major axis along the $y$-axis. The semi-major axis length is $\sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{2}$, and the semi-minor axis length is $\sqrt{5}$ along the $x$-axis.

2. Point on the Edge of the Shadow: The problem states that the point $(-5, 0)$ lies on the edge of the shadow, which matches the extent of the ellipse along the $x$-axis.

3. Line of Sight to the Lamp: The lamp is located at a point $(3, h)$, where $h$ is the height above the x-axis we need to determine. A line from the origin through $(-5, 0)$ and the point $(3, h)$ would represent the edge of the shadow.

4. Slope Calculation: The slope of the line from $(-5, 0)$ to $(3, h)$ is: $\text{slope} = \frac{h - 0}{3 - (-5)} = \frac{h}{8}$

5. Aligning with the Semi-Minor Axis: Since this line creates a boundary for the shadow cast by the lamp, it corresponds to a tangent line to the ellipse. Using geometry and similarity of triangles, it implies that: [ h =