= Solution
The <protein> fluctuation equation gives $\widetilde{\delta P}=(k_P\widetilde{\delta R}+\widetilde\eta_P)/(\gamma_P-i\omega)$. With the stated independent noises,
$$
S_P(\omega)=\frac{q_P}{\gamma_P^2+\omega^2}
+\frac{k_P^2q_R}{(\gamma_P^2+\omega^2)(\gamma_R^2+\omega^2)}.
$$
Partial fractions and the supplied integral give the general <variance>
$$
\boxed{\langle\delta P^2\rangle=\frac{q_P}{2\gamma_P}
+\frac{k_P^2q_R}{2\gamma_R\gamma_P(\gamma_R+\gamma_P)}.}
$$
The specific count-noise formula in the question additionally uses the usual steady birth-and-death noise strengths $q_R=2\gamma_RR_*$ and $q_P=2\gamma_PP_*$. Then $P_*=k_PR_*/\gamma_P$ reduces it to $P_*[1+k_P/(\gamma_R+\gamma_P)]$. Arbitrary unspecified $q_R,q_P$ do not by themselves imply this particular <Fano factor>.
The requested rewriting is
$$
\boxed{F=1+\frac{b}{1+\phi},\qquad b=\frac{k_P}{\gamma_R},\qquad
\phi=\frac{\gamma_P}{\gamma_R}=\frac{\tau_R}{\tau_P}.}
$$
Here $b$ is the <translational burst size>, the expected number of <proteins> produced during one <RNA>'s <mean> lifetime. The ratio $\phi$ compares removal rates, equivalently <RNA> to <protein> <mean> lifetimes. Poisson molecule-number statistics have <variance> equal to <mean>, hence $F=1$.
When <RNA> is much less stable than <protein>, $\phi\ll1$ and $F\simeq1+b$. At fixed degradation rates it rises linearly with the rate of <biological translation>, with exact slope $1/(\gamma_R+\gamma_P)$. The formal intercept at zero rate is the limit from positive <protein> <mean>; at exactly zero production both <mean> and <variance> vanish and their ratio is undefined. A larger <Fano factor> need not imply a larger relative <variance> $\operatorname{var}P/P_*^2$, because the <mean> also increases.
\Image[/past-exam-of-the-flows-fluctuations-and-complexity-course-of-the-university-of-oxford/2016-gene-expression-fano-factor.png]
{title=Protein Fano factor versus dimensionless rate of <biological translation> for a short-lived RNA and stable protein: exact two-stage result, burst approximation, and the Poisson baseline}
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