To simplify the expression ((a-\frac{a^2}{a+1}) \times \frac{a^2-1}{a^2+2a}), we can start by factoring the numerator and denominator of individual fractions.

Numerator of the first fraction:
(a - \frac{a^2}{a+1} = a - \frac{a^2}{a+1} = \frac{a(a+1) - a^2}{a+1} = \frac{a^2 + a - a^2}{a + 1} = \frac{a}{a + 1})

Denominator of the first fraction:
(a^2 + 2a = a(a+2))

Numerator of the second fraction:
(a^2 - 1 = (a+1)(a-1))

Denominator of the second fraction:
(a^2 + 2a = a(a+2))

Putting these back into the expression, we get:
(\frac{\frac{a}{a + 1} \times (a+1)(a-1)}{a(a+2)} = \frac{a(a-1)}{a(a+2)} = \frac{a-1}{a+2})

Therefore, ((a-\frac{a^2}{a+1}) \times \frac{a^2-1}{a^2+2a} = \frac{a-1}{a+2}).