If 30 cm sq (should be written sm^2 with 2 as a superscript) is the area, squaring it is meaningless. A "square centimeter" squared creates a unit of a nonexistent quantity, both geometrically and physically. Pay attention to dimensions of units when checking correctness of your solution.

To complete the solution leading to the first answer use some algebra -- not an uncommon approach in geometry!

Let the the length of the height of the triangle be H (keep in mind that this is cm). Then the length of the base is H+4 (also cm).

As a major geometric fact, the area of a triangle is half of the base length times the height length, that is H(H+4)/2 (the dimension of this product is cm^2, or sq cm).

According to the condition of the problem, the area is equal to 30 cm^2, so we get

H(H+4)/2=30, or H^2+4H-60=0. Solve the quadratic equation; you'll get one root equal to 6 (this is your height), and another root negative - it means nothing, discard it (can you explain where does it come from?). 

Remember that all quantities are either in cm or in cm^2, and the dimension of the left side of the equation is equal to that of the right side.