To solve the inequality, we first need to expand the right side of the inequality:

(x + 3)(x + 1) = x^2 + x + 3x + 3 = x^2 + 4x + 3

Now, our inequality becomes:

x^2 + 4x > x^2 + 4x + 3

Next, we can simplify the inequality:

x^2 + 4x > x^2 + 4x + 3

Subtract x^2 and 4x from both sides:

0 > 3

Since this inequality is always false (0 is not greater than 3), the solution to the original inequality x^2 + 4x > (x + 3)(x + 1) is all real numbers.