Answer
$x=2$
Work Step by Step
Note that $4=2^2$ so the given equation is equivalent to:
$2^{3x}=4^{x+1}$
$2^{3x}=(2^2)^{x+1}$
$2^{3x}=2^{2(x+1)}$
$2^{3x}=2^{2x+2}$
Using the rule $a^m=a^n\longrightarrow m=n$ gives:
$3x=2x+2$
$3x-2x=2$
$x=2$
Check:
$2^{3*2}=4^{2+1}$
$2^{6}=4^{3}$
$64=64$
Thus, the answer is correct.