Can you really multiply any two digit numbers in seconds using this viral mental math trick explain the algebra

This trick is all over Instagram:

To multiply $23 \times 47$:

Total: $800 + 140 + 120 + 21 = 1081$

But isnt this just FOIL? $(20+3)(40+7)$ expanded? So its not really a "trick" but just basic algebra?

I showed this to my friend and she was amazed but I was like "thats just the distributive property."

Actually theres another trick: to square a number ending in 5 like $75^2$:

Append 25: $5625$

This one actually IS surprising. Why does this work? Prove it with algebra.