Please bear with me my maths is very rusty.
If we have $x^2 + x^2$ this should be $2x^2$ meaning if powers are the same we can do the addition.
The question I have is how is $2^{x+1} + 2^{x+1} = 2.2^{x+1}$ then they apply the exponent rule and the final result is $2^{x+2}$.
The exponent rule makes sense to me since they are now multiplying and $2^1 * 2^{x+1}$ but why isnt $2^{x+1} + 2^{x+1} = 4^{x+1}$ since $4^{x+1} = 2.2^{x+1}$ but $4^{x+1} \neq2^{x+2} $
Your mistake is not uncommon. What you've done is ignore the invisible brackets.
When you break down the exponent, you really have $$4^{x+1}=(2×2)^{x+1}=2^{x+1}×2^{x+1}\neq 2×2^{x+1}$$
with the final part as you observed. This is because the exponent applies to the entire base, and in order to maintain the same expression, you have to keep the factors of that base together as an object. In mathematics, of course, that means putting in a pair of brackets; they weren't written originally because we can already see $4$ is a single object.