The equation $$x^2 - 4x + [x] + 3 = x[x]$$, where $$[x]$$ denotes the greatest integer function, has:

We need to solve $$x^2 - 4x + [x] + 3 = x[x]$$, where $$[x]$$ is the greatest integer function.

We start by rearranging the equation.

$$(x^2 - 4x + 3) = x[x] - [x] = [x](x - 1)$$

$$(x - 1)(x - 3) = [x](x - 1)$$

Next, we analyze cases based on the factor $$(x - 1)$$.

Case 1: $$x = 1$$.

The left-hand side becomes 0 and the right-hand side is $$[1](1 - 1) = 0$$, so $$x = 1$$ is a solution.

Case 2: $$x \neq 1$$.

Dividing both sides of $$(x - 1)(x - 3) = [x](x - 1)$$ by $$(x - 1)$$ gives

$$x - 3 = [x]$$

Since $$[x]$$ is an integer, $$x - 3$$ must be an integer, implying $$x$$ is an integer.

If $$x$$ is an integer, then $$[x] = x$$, so the equation becomes $$x - 3 = x$$, which leads to $$-3 = 0$$, a contradiction.

Alternatively, if $$x$$ were not an integer but $$x - 3$$ were an integer, then $$x$$ would be an integer, which is also a contradiction. Hence no solutions arise for $$x \neq 1$$.

Therefore, the only solution to the equation is $$x = 1$$.

Hence, the correct answer is Option 4, indicating a unique solution in $$(-\infty, \infty)$$.