To find the sum of the polynomial functions $f(x) = 3x^2 - 5x + 7$ and $g(x) = x^2 - 4x - 3$, use the definition $(f + g)(x) = f(x) + g(x)$. Substitute the given expressions to get $(3x^2 - 5x + 7) + (x^2 - 4x - 3)$. Rewrite without parentheses: $3x^2 - 5x + 7 + x^2 - 4x - 3$. Group like terms together: $3x^2 + x^2 - 5x - 4x + 7 - 3$. Combine the like terms to find the resulting polynomial function: $(f + g)(x) = 4x^2 - 9x + 4$.