Possible Integer Side Lengths of Quadrilateral

By Generalized Polygon Inequality, the length of any side of a polygon must be strictly less than the sum of the lengths of all other sides.

Let the fourth side be $x$ (an integer).

1. $x < 1 + 2 + 4 \implies x < 7$
2. $4 < 1 + 2 + x \implies x > 1$

Combining these conditions: $1 < x < 7$

Since $x$ is an integer, the possible values of $x$ are: $x \in \{2, 3, 4, 5, 6\}$ There are 5 possible integer lengths.