To find the difference of the polynomial functions $f(x) = 3x^2 - 5x + 7$ and $g(x) = x^2 - 4x - 3$, apply the definition $(f - g)(x) = f(x) - g(x)$. Substitute the polynomial expressions: $(3x^2 - 5x + 7) - (x^2 - 4x - 3)$. Distribute the subtraction by changing the sign of every term in the second polynomial: $3x^2 - 5x + 7 - x^2 + 4x + 3$. Group like terms together: $3x^2 - x^2 - 5x + 4x + 7 + 3$. Combine the like terms to find the resulting polynomial function: $(f - g)(x) = 2x^2 - x + 10$.