The larger leg of a right triangle is 2 feet longer than its smaller leg. the hypotenuse is 4 feet longer than the smaller leg. how many feet long is the larger leg?

We are given a right triangle whose legs we name as x, x+2 and x+4. The Pythagorean theorem states that the square of the hypotenuse's length is equal to sum of the squares of its legs. hence, we apply this to the given data:

a2 + b2 = c2
We substitute the given data
x2 + (x+2)2 = (x+4)2 
We expound the given squares
x2 + x2+ 4x + 4 = x2 + 8x +16
We group next the like terms
2x2-x2 = 8x -4x + 16 -4
x2 = 4x + 12
(x2-4x-12) = 0
we factor the given quadratic equation
(x-6)*(x+2) = 0
x1 = 6
x2 = -2

The valid solution set would be  6 since there are no negative measurements, so the answer to this problem would be 6+2 = 8 units.