<s>[INST] Prove the Inequality $\frac{1}{1-x}-\frac{x(3-x)(2-x)(13x^4-50x^3+89x^2-84x+36)}{4(1-x)(2x(1-x))^2}<1$ Can anyone suggest any hints to prove the following inequality:
$$\frac{1}{1-x} - \frac{x(3-x)(2-x)(13x^4 - 50x^3 + 89x^2 - 84x + 36)}{4(1-x)(2x(1-x))^2} < 1,$$
for all $x \in (0,1)$?
 [/INST] $$\frac{1}{1-x} - \frac{x(3-x)(2-x)(13x^4 - 50x^3 + 89x^2 - 84x + 36)}{4(1-x)(2x(1-x))^2} < 1,$$
$$\frac{x(3-x)(2-x)(13x^4 - 50x^3 + 89x^2 - 84x + 36)}{4(1-x)(2x(1-x))^2} > \frac{1}{1-x}-1,$$
$$\frac{x(3-x)(2-x)(13x^4 - 50x^3 + 89x^2 - 84x + 36)}{4(1-x)(2x(1-x))^2} > \frac{x}{1-x}.$$
For $x\in(0,1)$ we can prove that
$$\frac{(3-x)(2-x)(13x^4 - 50x^3 + 89x^2 - 84x + 36)}{4(2x(1-x))^2} > 1,$$
or
$$(3-x)(2-x)(13x^4 - 50x^3 + 89x^2 - 84x + 36) -  4(2x(1-x))^2 >0.$$
Taking in account that for $x\in(0,1)$
$$x(1-x)\leq \dfrac14,\quad 3-x>2,\quad 2-x>1,$$
$$13x^4 - 50x^3 + 89x^2 - 84x + 36 = 0.5x^4 + 12.5(x-1)^4 +\dfrac1{14}(14x-17)^2+\dfrac{20}7\geq \dfrac{20}7,$$
easy to prove the required inequality.
</s>