$A=a : \frac{a-1}{2} - \frac{a^{3} -3a(a-1)}{2a^{2} + 2a} .\frac{-4a}{a^{2} + 1-2a} - \frac{4a^{2}}{a^{2} - 1} (ĐKXĐ: a \neq 0; a\neq ±1)$
$= a.\dfrac{2}{a-1} - \dfrac{a^{3} - 3a(a-1)}{2a(a+1)}.\dfrac{-4a}{(a-1)^{2}} - \dfrac{4a^{2}}{(a-1)(a+1)}$
$=\dfrac{2a}{a-1} - \dfrac{(a^{3} - 3a^{2} + 3a).(-4a)}{2a(a+1).(a-1)^{2}} - \dfrac{4a^{2}}{(a-1)(a+1)}$
$=\bigg(\dfrac{2a(a+1)}{(a-1)(a+1)} - \dfrac{(2a)^{2}}{(a-1)(a+1)} \bigg) - \dfrac{-2a(a^{2} - 3a + 1)}{(a+1)(a-1)^{2}}$
$=\dfrac{2a(a+1) - 2a^{2}}{(a-1)(a+1)} - \dfrac{-2a(a^{2} - 3a + 1)}{(a+1)(a-1)^{2}}$
$=\dfrac{2a(a + 1 - 2a)}{(a-1)(a+1)} - \dfrac{-2a(a^{2} - 3a + 1)}{(a+1)(a-1)^{2}}$
$=\dfrac{2a(1-a)}{(a-1)(a+1)} - \dfrac{-2a(a^{2} - 3a + 1)}{(a+1)(a-1)^{2}}$
$=\dfrac{-2a(a-1)^{2} + 2a(a^{2} - 3a + 1)}{(a+1)(a-1)^{2}}$
$=\dfrac{2a[-(a-1)^{2} + a^{2} - 3a + 1]}{(a+1)(a-1)^{2}}$
$=\dfrac{2a[-a^{2} + 2a - 1 + a^{2} - 3a + 1]}{(a+1)(a-1)^{2}}$
$=\dfrac{2a(-a)}{(a+1)(a-1)^{2}}$
$=-\dfrac{2a^{2}}{(a+1)(a-1)^{2}}$
