How to Solve $3^{x+2} + 2^{x+2} + 2^x = 2^{x+5} + 3^x$

Question

I was wondering how to solve this equation for x
$3^{x+2} + 2^{x+2} + 2^x = 2^{x+5} + 3^x$
This equation was found on the CEMC eWorkshop for Euclid, but no solution was included.

Thank you, Eddie

Answer 1

$$3^{x+2} + 2^{x+2} + 2^x = 2^{x+5} + 3^x$$

$$9\cdot 3^x +4\cdot 2^x+2^x=32\cdot 2^x+3^x$$

$$8\cdot 3^x=27\cdot 2^x$$ $$\left(\frac{3}{2}\right)^x=\frac{27}8=\left(\frac{3}{2}\right)^3$$

Hence $x=3$.