<s>[INST] Find the solutions of the equation $\frac{a^2(x-b)(x-c)}{(a-b)(a-c)}+\frac{b^2(x-a)(x-c)}{(b-a)(b-c)}+\frac{c^2(x-a)(x-b)}{(c-a)(c-b)}=3x-2$ 
Find the solutions of the equation $$\frac{a^2(x-b)(x-c)}{(a-b)(a-c)}+\frac{b^2(x-a)(x-c)}{(b-a)(b-c)}+\frac{c^2(x-a)(x-b)}{(c-a)(c-b)}=3x-2$$

If we consider the expression as an identity and substitute $x=a$ we get the value of $a=1,2$ similarly $b=1,2$ and $c=1,2$
If we substitute these values into the expression at least one of the denominator will be equal to $0$ which is not possible therefore this expression is not an identity. How to proceed further?
 [/INST] Add $-x^2$ on both sides , Let expression on LHS be $f(x)$
Therefore LHS will be a quadratic in $x$ will be become zero three times clearly as
$f(a)=f(b)=f(c)=0$ Now if a quadratic equation has more than three roots, then all complex numbers will be it's roots and therefore it will identically be equal to zero.
$f(x)\equiv 0$ and therefore since $f(x)=3x-2-x^2$
$3x-2-x^2\equiv 0$
which is possible for $x=1,2$ only
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