Express all terms in the equation with a common base, which is 2.

$$4 \times 2^x = \left(\frac{1}{8}\right)^{2x-3}$$

We know that $4 = 2^2$ and $\frac{1}{8} = 8^{-1} = (2^3)^{-1} = 2^{-3}$.
Substitute these into the equation:

$$2^2 \times 2^x = (2^{-3})^{2x-3}$$

$$2+x = -6x + 9$$

Add $6x$ to both sides and subtract 2 from both sides:

$$x + 6x = 9 - 2$$ $$7x = 7$$

Divide by 7:

$$x = \frac{7}{7}$$ $$x = \boxed{\text{1}}$$