To simplify the expression M, let's break it down step by step:

M = (sqrt(a)/2 - 1/2sqrt(a)) * (a - sqrt(a)/sqrt(a+1) - a + sqrt(a)/sqrt(a-1))

First, let's simplify the first part of the expression:

(sqrt(a)/2 - 1/2sqrt(a)) = (sqrt(a) - 1)/2sqrt(a)

Now, let's simplify the second part of the expression:

(a - sqrt(a)/sqrt(a+1) - a + sqrt(a)/sqrt(a-1)) = (sqrt(a)/sqrt(a+1) - sqrt(a)/sqrt(a-1))

To simplify this further, we need to find a common denominator for the two terms:

(sqrt(a)/sqrt(a+1) - sqrt(a)/sqrt(a-1)) = (sqrt(a)(sqrt(a-1)) - sqrt(a)(sqrt(a+1)))/(sqrt(a+1)sqrt(a-1))

Now, let's simplify the numerator:

(sqrt(a)(sqrt(a-1)) - sqrt(a)(sqrt(a+1))) = sqrt(a)(sqrt(a-1) - sqrt(a+1))

Finally, let's simplify the denominator:

(sqrt(a+1)sqrt(a-1)) = sqrt((a+1)(a-1)) = sqrt(a^2 - 1)

Putting it all together, the simplified expression for M is:

M = (sqrt(a) - 1)/2sqrt(a) * sqrt(a)(sqrt(a-1) - sqrt(a+1))/sqrt(a^2 - 1)

Note: It's important to double-check the expression and make sure there are no errors in the original equation before simplifying.