To simplify the expression, we can start by combining the fractions in the numerator:

P = (1/√x + √x)/(√x + 1) : √x/(x + √x)

Next, we can simplify the denominator by multiplying both the numerator and denominator by the conjugate of the denominator:

P = (1/√x + √x)/(√x + 1) * (√x - 1)/(√x - 1) : √x/(x + √x)

This gives us:

P = [(1 - √x)/(x)] * (√x - 1)/(√x + 1) : √x/(x + √x)

Now, we can simplify further by canceling out common factors:

P = [(1 - √x)(√x - 1)]/(x(√x + 1)) : √x/(x + √x)

Expanding the numerator:

P = (1 - 2√x + x)/(x(√x + 1)) : √x/(x + √x)

Now, we can simplify the expression by multiplying both the numerator and denominator by (√x + 1):

P = (1 - 2√x + x)(√x + 1)/(x(√x + 1)(√x + 1)) : √x/(x + √x)

Simplifying further:

P = (1 - 2√x + x)(√x + 1)/(x(√x + 1)^2) : √x/(x + √x)

Expanding the denominator:

P = (1 - 2√x + x)(√x + 1)/(x(x + 2√x + x)) : √x/(x + √x)

Simplifying the denominator:

P = (1 - 2√x + x)(√x + 1)/(x(2x + 2√x)) : √x/(x + √x)

Now, we can cancel out common factors:

P = (1 - 2√x + x)/(2x + 2√x)

Simplifying further:

P = (x - 2√x + 1)/(2x + 2√x)

Therefore, the simplified expression is (x - 2√x + 1)/(2x + 2√x).