Solve (Sqrt4x+1/x+3)-(sqrtx-2/x+3)=1

Assuming the usual carelessness with parentheses, I'll go with

√((4x+1)/(x+3)) - √((x-2)/(x+3)) = 1
clearing the fractions, we have
√(4x+1) - √(x-2) = √(x+3)
now square both sides to get
4x+1 - 2√((4x+1)(x-2)) + x-2 = x+3
√((4x+1)(x-2)) = 2x-2
(4x+1)(x-2) = 4(x^2-2x+1)
4x^2 - 7x - 2 = 4x^2 - 8x + 4
x = 6
check:
√(25/9) - √(4/9) = 5/3 - 2/3 = 1