This cannot be factored using only rational expressions, 
but if you allow square roots in the factorization, then 
there are more than one way to do this, and each way gives
a different expression. 

Here's the first way:



Write a²+4a as its square root squared:



Then factor it as the difference of two squares:



Here's a second way:



Swap the second and third terms:



Factor out a - from the 2nd and 3rd terms:



Write the expression in parentheses as the 
square of its square root:



Then factor as the difference of two squares:




Here's a third way:



Add +4 after the 4a and subtract 4 after the -b^2



Factor the first three terms as a perfect square trinomial.
Factor a negative out of the last two terms:



Then write the expression in parentheses after the - sign 
as the square of its square root.



Then as



There are other ways as well, but they'll always involve
square roots.  

Edwin