The number of integral values of $k$ , for which $(x^{2} - x + 1) (k x^{2} - 3 k x - 5) < 0$ is

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Solution

$(x^{2} - x + 1) (k x^{2} - 3 k x - 5) < 0$
$x^{2} - x + 1 > 0$ as $D < 0$
When $k = 0$ , we get
$- 5 (x^{2} - x + 1) < 0 \Rightarrow x^{2} - x + 1 > 0$
Which is true.

Now,
$k x^{2} - 3 k x - 5 < 0$
Therefore,
$k < 0 \dots (1) D < 0 \Rightarrow 9 k^{2} + 20 k < 0 \Rightarrow k (9 k + 20) < 0 \Rightarrow k \in (- \frac{20}{9}, 0) \dots (2)$
From $(1)$ and $(2)$ ,we get
$\Rightarrow k \in (- \frac{20}{9}, 0)$

Hence, the integral values are $k = - 2, - 1, 0$

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