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Solution

If $P = \{4^n - 3n - 1 : n \in \mathbb{N}\}$ and $Q = \{9n - 9 : n \in \mathbb{N}\}$

Elements of P
$n=1: \quad 4^1 - 3(1) - 1 = 0$
$n=2: \quad 4^2 - 3(2) - 1 = 9$
$n=3: \quad 4^3 - 3(3) - 1 = 54$
$n=4: \quad 4^4 - 3(4) - 1 = 243$
$\Rightarrow P = \{0, 9, 54, 243, \ldots\}$

Elements of Q
$n=1: \quad 9(1) - 9 = 0$
$n=2: \quad 9(2) - 9 = 9$
$n=3: \quad 9(3) - 9 = 18$
$\Rightarrow Q = \{0, 9, 18, 27, \ldots\}$ = all multiples of 9

$P \subseteq Q$

$P \subseteq Q \Rightarrow P \cup Q = Q$

$\boxed{P \cup Q = Q}$