<s>[INST] Then find the value of ... Let $a, \: b, \: c$ be three variables which can take any value (Real or Complex). Given that $ab + bc + ca = \frac{1}{2}$; $a + b + c = 2$; $abc = 4$. Then find the value of $$\frac{1}{ab + c - 1} + \frac{1}{bc + a - 1} + \frac{1}{ac + b - 1}$$
I always get stuck in such problems. Please give a detailed solution and also give me an approach that I can use to solve other such kind of problems.
 [/INST] Another way is to find a cubic $p(x)$ with roots $ab+c, bc+a, ca+b$ (which is easier than the original problem) and then use simple transformations. Here using elementary symmetric polynomials and Vieta, we get $p(x) = x^3-\frac52x^2- \frac52x-\frac{65}4$.  
Now get the polynomial $p_1(x)$ which has roots $ab+c-1, bc+a-1, ca+b-1$, as $p_1(x) = p(x+1) = x^3+\frac12x^2-\frac92x-\frac{81}4$
Finally the polynomial with reciprocal roots is got by reversing coefficients just so:
$p_2(x) = -\frac{81}4x^3-\frac92x^2+\frac12x+1$
So the sum of these roots is just $-\frac92 \div \frac{81}4 = -\frac29$
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