$X$ is a random variable. $f_X(x)=2xdx$ is the density function of $X$ such that $x\in[0,1]$.$Y=\lfloor NX\rfloor$ such that $N\in \mathbb {N}$.
$Y=L$ if and only if $2\lfloor {NX} \rfloor=L$, not $\lfloor {2NX} \rfloor=L$.
I believe there is an error in your treatment of the floor function.
$\lfloor 2LX \rfloor \neq L$ for all $X > \frac{1}{2}$
Consider $X = \frac{3}{4}$ and $L = 8$:
$$\lfloor 2 \cdot 8 \cdot \frac{3}{4} \rfloor = \lfloor 12 \rfloor \neq 8$$
Does this help you work through the rest of the solution?