Solution:
[tex]A = \frac{2}{(a - b)} + \frac{2}{(b - c) } + \frac{2}{(c - a) } + \frac{(a-b)²+(b-c)²+(c-a)²}{(a - b)(b - c)(c - a)} \\ \: \: \: = \frac{2(b - c)(c - a) + 2(a - b)(c - a) + 2(a - b)(b - c)}{(a - b)(b - c)(c - a)} + \frac{(a-b)²+(b-c)²+(c-a)²}{(a - b)(b - c)(c - a)} \\ \: \: = \frac{2(bc - ab - {c}^{2} + ac) + 2(ac - {a}^{2} - bc + ab) + 2(ab - ac - {b}^{2} + bc) + a² - 2ab + b ² + b² - 2bc+c²+c²-2ac+a²}{(a - b)(b - c)(c - a)} \\ \: \: = \frac{2bc - 2ab - 2{c}^{2} + 2ac+ 2ac - 2 {a}^{2} - 2bc + 2ab+ 2ab - 2ac - 2 {b}^{2} + 2bc+ a² - 2ab + b ² + b² - 2bc+c²+c²-2ac+a²}{(a - b)(b - c)(c - a)} \\ \: \: = \frac{a² + b ² +c² }{(a - b)(b - c)(c - a)} \\ \: \: [/tex]
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