4 2x+1 -10.4 x +4=0

I think Guest is right:

This function doesn't ever appear to equal 0.....see the graph, here.........https://www.desmos.com/calculator/gbffzpgnyt

It appears that the graph has a horizontal asymptote at y = 4 and the function never crosses this.

Write that first term as $4\times 4^{2x}=4(4^{x})^{2}$ and see that you have a quadratic in $4^{x}.$

Replace $4^{x}$ by y (say) if it makes it easier to see.

I think that possibly desmos is interpreting that 10.4 in Chris's graph as 10 decimal point 4 rather than 10 times 4 ?

I had this one stashed away to come back to.  Well I am back !!

$4^{2x+1}-10*4^x+4=0\\ 4*4^{2x}-10*4^x+4=0\\~\\ Let\;m=4^x\\ 4*m^2-10*m+4=0\\ 2m^2-5m+2=0\\ 2m^2-4m-m+2=0\\ 2m(m-2)-1(m-2)=0\\ (2m-1)(m-2)=0\\ 2m-1=0\qquad or \qquad m-2=0\\ 2m=1\qquad or \qquad m=2\\ m=\frac{1}{2} \qquad or \qquad m=2\\~\\ $

$\frac{1}{2}=4^x \qquad or \qquad 2=4^x\\ 2^{-1}=2^{2x} \qquad or \qquad 2^1=2^{2x}\\ -1=2x \qquad or \qquad 1=2x\\ x=\frac{-1}{2} \qquad \quad or \qquad x=\frac{1}{2}\\$

Here is the graph

https://www.desmos.com/calculator/lbykd4ollu