If the equation x² + 5kx + 16 = 0 has no real roots, then :

1. k > $\dfrac{8}{5}$

2. k < $-\dfrac{8}{5}$

3. $-\dfrac{8}{5} \lt k \lt \dfrac{8}{5}$

4. none of these

Comparing x² + 5kx + 16 = 0 with ax² + bx + c = 0 we get,

a = 1, b = 5k and c = 16.

We know that,

Since equations has no real roots,

⇒ D < 0

⇒ b² - 4ac < 0

⇒ (5k)² - 4(1)(16) < 0

⇒ 25k² - 64 < 0

⇒ 25k² < 64

$\Rightarrow k^2 \lt \dfrac{64}{25} \\[1em] \Rightarrow |k| \lt \sqrt{\dfrac{64}{25}} \\[1em] \Rightarrow |k| \lt \dfrac{8}{5} \\[1em] \Rightarrow -\dfrac{8}{5} \lt k \lt \dfrac{8}{5}.$

Hence, option 3 is the correct option.