If the sum of the ages of a mother and her child is 63 now,
hence, 4 years ago this sum was 63-4-4 = 55.


Next, 484 = (mentally) 4*121 = 2*2*11*11.


So, we should combine these factors to make a sum 55 from them.
For example, this way works


       55 = 44 + 11 = (2*2*11) + 11,


so, we can assume that 4 years ago their age was 44 and 11;  
so now their ages are 44+4 = 48 (mother)  and 11+4 = 15.   ANSWER

Another way to solve it is to write this equation for their ages 4 years ago

    x*(55-x) = 484

    55x - x^2 = 484

    x^2 - 55x + 484 = 0


and then solve it using the quadratic formula or factoring, on your choice.