This is an exercise in "completing the square" for a quadratic equation. It's a handy trick that is useful for problems like this. Given the equation x^2 + bx + c, you can take the b coefficient, divide it by 2, square the result and then add and subtract the final result, like this:
f(x) = x^2 + bx + x = x^2 + bx + (b/2)^2 - (b/2)^2 + c = (x+(b/2))^2 -(b/2)^2 + c
So for your problem you have f(x)= x^2 + 6x + 4 = x^2 + 6x + 3^2 - 3^2 + 4 = (x+3)^2 -5.
That's part (a) done. Now rearrange to get it in the form x = .... You will end up with an equation that has a square root, which limits the range for f(x). And by the way, though in part (c) you wrote x >= -5 I think what you mean is f(x) >= -5.