A= [1/(x+2√x) - 1/(√x+1)] : (1-√x)/(x+4√x+4)

To simplify the expression, let's start by simplifying the numerator and denominator separately.

Numerator:
1/(x+2√x) - 1/(√x+1)

To simplify this, we need to find a common denominator. The common denominator is (x+2√x)(√x+1).

So, the expression becomes:
[(√x+1) - (x+2√x)] / [(x+2√x)(√x+1)]

Simplifying the numerator:
(√x+1) - (x+2√x)
= √x + 1 - x - 2√x
= (1 - x) - √x

So, the numerator simplifies to:
(1 - x) - √x

Denominator:
(1-√x)/(x+4√x+4)

To simplify this, we can multiply the numerator and denominator by the conjugate of the denominator, which is (x+4√x+4).

So, the expression becomes:
[(1-√x)(x+4√x+4)] / [(x+4√x+4)]

Expanding the numerator:
(1-√x)(x+4√x+4)
= x + 4√x + 4 - √x(x+4√x+4)
= x + 4√x + 4 - x√x - 4x - 4√x - 4
= -3x - x√x

So, the denominator simplifies to:
-3x - x√x

Now, we can rewrite the expression as:
[(1 - x) - √x] / [-3x - x√x]

Therefore, the simplified expression is:
[(1 - x) - √x] / [-3x - x√x]