It is quite possible this question has no answer -- that is, the area cannot be determined from the information given. It's a question I've created myself as I study for the GRE. No trigonometry is meant to be used, only basic logic about geometry.
(Please bear with me -- this GRE material is quite different than the formal math I am used to in the areas of analysis, abstract algebra, probability, etc.! -- I want to make sure my logic is sound and that I haven't made any leaps in logic or imposed any extra assumptions that weren't given!)
Please consider the figure below, not necessarily drawn to scale:
My question: can I determine the area of the triangle with the information given?
My idea: Draw a line from $B$ down to a new point $D$ on line $AC$ so that that it creates a right angle with $AC$:
Now, observe triangle $BCD$ above. By construction it is a right triangle, whose hypotenuse is length $5$. Therefore, it must be a "$3$-$4$-$5$" right triangle. (Is this step correct or have a made a jump??)
Therefore, I can determine $DC=3$ and $BD=4$.
Finally, I plug in these lengths to compute the area as $A=\frac{1}{2}\cdot 14 \cdot 4 =28$.
I decided to upgrade my comment into an answer.
Angle $C$ is free to vary. That would change the length of $BD$ while preserving the given lengths, which is were you went astray (the length of $BD$ is not necessarily $4$).
Now, in general, if we know two side lengths of a triangle (call them $x$ and $y$) and the angle measure between them (call it $\theta$), then the area of the triangle is given by $A = \frac{1}{2}xy\sin(\theta)$.
Applying this to our scenario, since the angle between the two given sides can vary, then so too can the area of the triangle. This lack of uniqueness is evident in the figure shown below: Every triangle shown has the prescribed side lengths, but all have different areas. In short, we cannot determine the area with the given information.