Question on how a "shortcut" works in math

Say you're given a problem such as:

77^2 - 73^2 = ?

Well, the "shortcut" that I learned for this is:

4(77 + 73) = 600

Thus,

77^2 - 73^2 = 600

But... what's the math behind that?

a^2 + b^2 = c
4(a + b) = c

So,

a^2 + b^2 = 4(a + b)

Any explanations how that works?

I'm having trouble figuring out what you are even saying your shortcut is
You describe the shortcut as being a^2 + b^2 = 4(a + b) but your example problem isn't in this form. (it has subtraction not addition)

It just seems to be pure chance that 77^2 - 73^2 = 4(77 + 73)

In most cases it won't be true that a^2 + b^2 = 4(a + b) nor will it true that a^2 - b^2 = 4(a + b)

A shortcut that will always work though is:
a^2 - b^2 = (a-b)(a+b)