Explanation:

To determine when the quadratic expression \(mx^2 + mx + 8x + 9\) is a perfect square, let's compare it to a perfect square trinomial \((ax + b)^2 = a^2x^2 + 2abx + b^2\):

- For \(mx^2 + mx + 8x + 9\) to be a perfect square:

1. The coefficient \(a^2\) must equal \(m\).

2. The term \(2abx\) should equal \(m + 8\) since it's the middle term in the trinomial.

3. Lastly, the constant term \(b^2\) should equal \(9\), giving us \(b = 3\) or \(-3\).

- Exploring the options:

- Option 1 (1, 4): Does not satisfy all conditions.

- Option 2 (4, 9): Solving gives \(a = 2, b = 3\). Fits conditions.

- Option 3 (9, 16): Solving gives \(a = 3, b = 3\). Fits conditions.

- Option 4 (4, 16): Matches for \(a = 2\) and \(b = 3\).

- Correct Options:

- Option 2: (4, 9)

- Option 4: (4, 16)