Answer
$x=\frac{2}{5}$
Work Step by Step
Note that $27=3^3$ so the give equation is equivalent to:
$3^{x+2}=27^{2x}$
$3^{x+2}=(3^3)^{2x}$
$3^{x+2}=3^{3\cdot 2x}$
$3^{x+2}=3^{6x}$
Using the rule $a^m=a^n\longrightarrow m=n$ gives:
$x+2=6x$
$2=6x-x$
$2=5x$
$x=\frac{2}{5}$
Check:
$3^{x+2}=27^{2x}$
$3^{2/5+2}=27^{2*2/5}$
$3^{12/5}=27^{4/5}$
$3^{12/5}=(3^3)^{4/5}$
$3^{12/5}=3^{3*4/5}$
$3^{12/5}=3^{12/5}$
Thus, the answer is correct.