$\dfrac{(8x^3 + 36x^2 + 54x + 27){(x-1)}^2}{(x^3 + 6x^2 + 12x + 8){(x+1)}^2} < 0$

Find all the possible values of $x$ .

$[1 , 3)$
$(-\infty , 4]$
$(-2 , -1.5)$
$[4 , \infty)$

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Relevant wiki: Wavy Curve Method$\begin{aligned} \dfrac{(8x^3 + 36x^2 + 54x + 27){(x-1)}^2}{(x^3 + 6x^2 + 12x + 8){(x+1)}^2} & < 0\\ \\ \dfrac{(8x^3 + 27 + 18x(2x + 3)){(x-1)}^2}{(x^3 + 8 + 6x(x + 2)){(x+1)}^2} & < 0\\ \\ \dfrac{({2x}^3 + {3}^3 + (3 × 2x × 3)(2x + 3)){(x-1)}^2}{(x^3 + 2^3 + (3 × x × 2)(x + 2)){(x+1)}^2} & < 0\\ \\ \dfrac{{(2x + 3)}^3{(x-1)}^2}{{(x + 2)}^3{(x+1)}^2} & < 0\\ \\ \text{Roots of odd powers} & = -2 , -1.5\\ \text{Roots of even powers} & = -1 , 1 \end{aligned}$

Since the inequality is only $<$ 0 and not $\geq$ 0 or $\leq$ 0, we dont care about the roots of the denominators.

$\therefore \boxed{x \in (-2 , -1.5)}$