Final answer:
The range of the function f(x) = -(x + 5)(x + 1) is all real numbers less than or equal to 4, found by determining the maximum value at the axis of symmetry, x = -3.
Step-by-step explanation:
The question asks about the range of the function f(x) = -(x + 5)(x + 1). To determine this, we must analyze the turning points and the end behavior of the function. The function is a parabola opening downwards since the leading coefficient (negative in this case) determines the direction of the parabola. The turning points, also known as the vertices, are found from the roots of the quadratic, which in this case are x = -5 and x = -1. The axis of symmetry is the mid-point between the roots, which is x = -3. At this point, f(x) achieves its maximum value.
By substituting x = -3 into the function, we find the maximum value of f(x): f(-3) = -(-3 + 5)(-3 + 1) = -2 * -2 = 4. Since the parabola opens downwards, the function will take on all values less than or equal to this maximum. Therefore, the range of the function is all real numbers less than or equal to 4, which is option a.