Given,

$\rm 2^{x} + 2^{x + 2} = 5$

By the law of indices, $\rm a^{m+n} = a^{m} \cdot a^{n}$,

$\rm or, 2^{x} + 2^{x} \cdot 2^{2} = 5$

$\rm or, 2^{x} \left ( 1 + 2^{2} \right ) = 5$

$\rm or, 2^{x} \cdot (1 + 4) = 5$

$\rm or, 2^{x} \cdot 5 = 5$

Dividing both sides of the equation by 5, we get,

$\rm or, 2^{x} \cdot \frac{5}{5} = \frac{5}{5}$

$\rm or, 2^{x} = 1$

$\rm or, 2^{x} = 2^{0}$

The bases of the terms on both sides of the equation are the same, so we equate their powers.

$\rm \therefore x = 0$

Hence, the required value of x is 0.