If $(2x+3):(5x+4)$ is the triplicate ratio of $3:4$, find the square of $x$.

1. The triplicate ratio of a ratio $a:b$ is defined as $a^3:b^3$.
$$\text{Triplicate ratio of }3:4=3^3:4^3=27:64$$

2. According to the question, set up the proportion equation:

$$\frac{2x+3}{5x+4}=\frac{27}{64}$$

3. Cross-multiply and solve for $x$:

$$64(2x+3)=27(5x+4)$$

$$128x+192=135x+108$$

$$135x-128x=192-108$$

$$7x=84⟹x=12$$

4. Find the square of $x$:

$$\text{Square of }x=x^2=12^2=144$$