Given

$\rm 3^{x} + 3^{x + 1} + 3^{x + 2} + 3^{x + 4} = 846$

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

$\rm or, 3^{x} + 3^{x} \cdot 3^{1} + 3^{x} \cdot 3^{2} + 3^{x} \cdot 3^{4} = 846$

$\rm or, 3^{x} \left ( 1 + 3^{1} + 3^{2} + 3^{4} \right ) = 846$

$\rm or, 3^{x} \left ( 1 + 3 + 9 + 81 \right ) = 846$

$\rm or, 3^{x} \left ( 94 \right ) = 846$

Dividing both sides of the equation by 94, we get,

$\rm or, 3^{x} \frac{94}{94} = \frac{846}{94}$

$\rm or, 3^{x} = 9$

$\rm or, 3^{x} = 3^{2}$

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

$\rm \therefore x = 2$

Hence, the required value of x is 2.