Step 1:

$X\sim P(4)$

$\therefore P(X=x)=\large\frac{e^{\Large -4}4^{\Large x}}{x!}$$\qquad x=0,1,2....$

Step 2:

$P(2 \leq X <5)=P(X=2)+P(X=3)+P(X=4)$

$\qquad\qquad\quad\;\;=e^{-4}\big[\large\frac{4^2}{2!}+\frac{4^3}{3!}+\frac{4^4}{4!}\big]$

$\qquad\qquad\quad\;\;=0.0183\big[8+\large\frac{32}{3}+\frac{32}{3}\big]$

$\qquad\qquad\quad\;\;=0.0061[88]$

$\qquad\qquad\quad\;\;=0.5368$