The dominant term is $x^4$ (Imagine that you expanded both the numerator and denominator. What would the highest power of $x$ be?). So, you could either multiply numerator and denominator by $1/x^4$ (see below), or factor as follows: $\eqalign{ {(2x-5)^4 \over (2x^2+1)(3x^2-2)} &={ \bigl(x(2-{5\over x})\bigr)^4\over x^2 (2+{1\over x^2}) \cdot x^2(3-{2\over x^2 } ) }\cr &={x^4 (2-{5\over x})^4 \over x^2(2+{1\over x^2})\cdot x^2(3-{2\over x^2 })}\cr &={(2-{5\over x})^4\over (2+{1\over x^2})(3-{2\over x^2})}. } $
The limit as $x$ tends to infinity is $ {2^4\over 2\cdot 3}=16/6 =8/3.$
The $(2-5/x)^2$ is probably a typo..
Using the other approach, where you multiply numerator and denominator by $1/x^4$:
In the numerator, to distribute $1/x^4$ over $(2x-5)^4$: $\textstyle {1\over x^4}(2x-5)^4 = \bigl( {1\over x} (2x-5) \bigr)^4=(2-{5\over x})^4. $ (just using $a^nb^n=(ab)^n$ here).
In the denominator, you could do the following: $\textstyle {1\over x^4}(2x^2+1)(3x^2-2) ={1\over x^2}(2x^2+1) {1\over x^2}(3x^2-2)= (2+{1\over x^2})(3-{2\over x^2}) $ (or just expand $(2x^2+1)(3x^2-2)$ first and then distribute the $1/x^4$ across).