Your error was in going from
$\displaystyle 2^x2^x2^{-1}=2^4+(2)2^x$ to $\displaystyle 2^x2^x=2^5+(4)2^x$
You inadvertently switched the addition and multiplication symbols on the right-hand side.
You could also try
$\displaystyle 2^{2x-1}-2^{x+1}=2^4$
$\displaystyle 2^{2x}-2^{x+2}=2^5$
$\displaystyle 2^x\left[2^x-4\right]=32$
The only two positive numbers that differ by 4 and whose product is 32 are 8 and 4 so by inspection $\displaystyle 2^x=8$
Hello, PythagorasNeophyte!
Sheesh . . . you're at this level of algebra
. . and you're still using $\displaystyle \times$ for multiplication?
$\displaystyle 2^{2x-1} \:=\:16 +2^{x+1}$
We have: .$\displaystyle 2^{2x-1} - 2^{x+1} - 16 \:=\:0$
. . . . . $\displaystyle 2^{2x}\cdot2^{-1} - 2^x\cdot 2 - 16 \:=\:0 $
. . . . . . . $\displaystyle \frac{1}{2}\!\cdot\!2^{2x} - 2\cdot2^x - 16 \:=\:0$
Multiply by 2: .$\displaystyle 2^{2x} - 4\cdot2^x - 32 \:=\:0$
Let $\displaystyle y = 2^x\!:\;\;y^2 - 4y - 32 \:=\:0\quad\Rightarrow\quad (y + 4)(y-8) \:=\:0 \quad\Rightarrow\quad y \:=\:-4,\:8$
Back-substitute: .$\displaystyle \begin{Bmatrix} 2^x \:=\:\text{-}4 & \Rightarrow & \text{no real roots} \\
2^x \:=\:8 & \Rightarrow & \boxed{x \:=\:3} \end{Bmatrix}$
If you want a fancy solution
instead of a quadratic....
$\displaystyle 2^{2x-1}=2^4+2^{x+1}$
divide both sides by 2 to obtain $\displaystyle 2^x$ on the right
$\displaystyle 2^{2x-2}=2^3+2^x$
$\displaystyle 2^{2x-2}-2^3=2^x$
Divide both sides by $\displaystyle 2^x$
$\displaystyle 2^{x-2}-2^{3-x}=1$
The only powers of 2 whose index moduli and values differ by 1 are $\displaystyle 2^0=1$ and $\displaystyle 2^1=2$
$\displaystyle \Rightarrow\ x=3$