Factor each denominator:
x^2 + 2x + 1 = (x+1)^2
x^2 + 4x + 3 = (x+3)(x+1)
Now take the common and non common factors with the highest exponents and multiply them.
(x+3).(x+1)^2 is the least common denominator.
If you expand that product you will get x^3 + 5x^2 + 7x +3
