It Looks Easy as it is

We note that $4^{x-1} = \frac{4^{x}}{4}$ .

Let $p = 4^{x}$ , then the equation becomes

$\frac{p}{4} + p = \frac{15}{4}$

$p + 4p = 15 \Rightarrow p = 3$ .

We have $4^{x} = 2^{2x} = 3$ .

$2^{2x} = 3 \Rightarrow 2^{x} = \sqrt{3}$ .

Raising both sides, we have

$2^{3x} = (\sqrt{3})^{3}$

$8^{x} = 3\sqrt{3} = a\sqrt{a}$ .

$a=3$ .

Hence, $4^{3} \cdot 3^{4} = \boxed{5184}$