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 = x^{2} + 5x

Moving the 104 over...

x^{2} + 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.