<s>[INST] If $t = \frac{(2-x)(3-x)}{3x^2 -7x + 6 }$, then find $\frac{-x^4 + 2x^3 - 3x^2 + 2x}{3x^2 -7x + 6 }$? Suppose that $t =\frac{(2-x)(3-x)}{3x^2 -7x + 6 }$.  Can you, please, give me any hints to express the following in terms of $t$:
$$\frac{-x^4 + 2x^3 - 3x^2 + 2x}{3x^2 -7x + 6 }?$$
 [/INST] Cross multiply the definition of $t$ to get $t(3x^2-7x+6) = (x^2-5x+6)$.  Write the whole business as a polynomial in $x$:  $(3t-1)x^2 +(-7t+5)x +(6t-6) = 0$.  Use the quadratic formula to get $x= \frac{7t-5 \pm\sqrt{(-7t+5)^2 - 4(3t-1)(6t-6)}}{2(3t-1)}$ and simplify a little.  Plug this monstrosity in for $x$ in your expression and may God have mercy on your soul.
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