So you have a right triangle. The base can be called 'x', and the height can be called 'x+5', since it's 5 units longer than the base. The formula for the area of a triangle is
A = (1/2)(b)(h)
so we can rewrite that using our information as:
52 = (1/2)(x)*(x+5)
Moving the one-half over, we get:
104 = (x)*(x+5)
If we expand the right-half of the equation, it becomes:
104 = x2 + 5x
Moving the 104 over...
x2 + 5x - 104 = 0
This is where factoring comes in handy. 13 and -8 are the numbers we'll use, since 13 - 8 = 5, our b term, and 13 * -8 = -104, our c term. Recall that for factoring, we need two numbers whose sum makes the b term, and product makes the c term. We can simplify this expression to:
(x+13)*(x-8) = 0
What numbers, when substituted for x, will make the expression zero? -13 and +8 will, so those are your possible values for x. -13 doesn't make sense since you can't have a negative length for your sail, so the answer must be 8ft long for the base, and 8+5ft long for the height.