I haven't submitted this question to the normal, rigorous fact-checking of a typical Radical Reference interaction, but I did find a scholarly journal article with the history of two plus two by Houston Euler. It's from Mathematics Magazine, Vol. 63, No. 5. (Dec., 1990), pp. 338-339. If you have access to JSTOR, here's a stable URL.

You have to start with 1 = 2.

let a = b
a² = ab (Multiply both sides by a)
a² + a² - 2ab = ab + a² - 2ab (Add {a² - 2ab} to both sides)
2(a² - ab) = a² - ab (Factor the left, and collect like terms on the right)
2 = 1 (Divide both sides by {a² - ab})

Once you've got that settled, you can start on 2 + 2 = 5:

2 = 1 + 1, so therefore

x = 2 + 2 goes to

x = 2 + (1 + 1)

Because you now have proof that 1 = 2

x = 2 + (1 + 2)

x = 2 + 3

x = 5

Substituting the original equation for x:

2 + 2 = 5

Thanks to members of the LightningBug development team for assistance on this question.