As we know that, a dimensionless quantity will have dimensional formula as $\left[\mathrm{M}^0 \mathrm{~L}^0 \mathrm{~T}^0\right]$.
As given that expression is $P=E l^2 m^{-5} G^{-2}$ (since $E, L, G$ have dimensional formulas) where
$\mathrm{E}$ is energy $[E]=\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right]$
$m$ is $\operatorname{mass}[m]=[\mathrm{M}]$
$L$ is angular momentum $[L]=\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]$
$G$ is gravitational constant $[G]=\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]$
Substituting dimensions of each term in the given expression,
$$
\begin{aligned}
&{[P]=\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right] \times\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]^2 \times[\mathrm{M}]^{-5}} \\
&\times\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]^{-2} \\
&=\left[\mathrm{M}^{1+2-5+2} \mathrm{~L}^{2+4-6} \mathrm{~T}^{-2-2+4}\right]=\left[\mathrm{M}^0 \mathrm{~L}^0 \mathrm{~T}^0\right] \\
&
\end{aligned}
$$
Thus, $P$ is a dimensionless quantity.