The problem asks you to find $g(x)$, given that:

$f \circ g(x) = 2x^2 - 4x + 3 \quad \text{and} \quad f(x) = 2x - 1$

Step-by-step solution:

1. The composition $f \circ g(x)$ means that: $f(g(x)) = 2x^2 - 4x + 3$
2. Since $f(x) = 2x - 1$, we replace $f(g(x))$ with $2g(x) - 1$: $2g(x) - 1 = 2x^2 - 4x + 3$
3. Now, solve for $g(x)$: $2g(x) = 2x^2 - 4x + 3 + 1$ $2g(x) = 2x^2 - 4x + 4$
4. Divide both sides by 2: $g(x) = x^2 - 2x + 2$

Thus, $g(x) = x^2 - 2x + 2$.

Would you like further details or explanations? Here are some additional questions to explore this topic:

1. What is the general process for solving a composite function problem?
2. How do you verify the solution of a composite function problem like this one?
3. Can you explain the difference between $f(g(x))$ and $g(f(x))$?
4. What would the process look like if the function $f(x)$ were more complex?
5. How can solving composite functions help in real-world applications?

Tip: When working with composite functions, make sure you substitute the inner function correctly into the outer one to avoid errors.