(i) ` 2P^(2)+9P+9 = 0`
(ii) `15Q^(2) + 16Q + 4 = 0`

Options

A. `P gt Q`
B. `P lt Q`
C. `P ge Q`
D. ` P le Q`

Explanation

To solve the equations given in the question, we will follow these steps:

### Step 1: Solve the first equation `2P^2 + 9P + 9 = 0`

1. **Identify the coefficients**: Here, \(a = 2\), \(b = 9\), and \(c = 9\).
2. **Calculate the discriminant**:
\[ D = b^2 - 4ac = 9^2 - 4 \times 2 \times 9 = 81 - 72 = 9 \]
3. **Since the discriminant is positive, we will have two real and distinct roots**.
4. **Use the quadratic formula**:
\[ P = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{9}}{2 \times 2} = \frac{-9 \pm 3}{4} \]
5. **Calculate the two roots**:
- First root:
\[ P_1 = \frac{-9 + 3}{4} = \frac{-6}{4} = -\frac{3}{2} \]
- Second root:
\[ P_2 = \frac{-9 - 3}{4} = \frac{-12}{4} = -3 \]

### Step 2: Solve the second equation `15Q^2 + 16Q + 4 = 0`

1. **Identify the coefficients**: Here, \(a = 15\), \(b = 16\), and \(c = 4\).
2. **Calculate the discriminant**:
\[ D = b^2 - 4ac = 16^2 - 4 \times 15 \times 4 = 256 - 240 = 16 \]
3. **Since the discriminant is positive, we will have two real and distinct roots**.
4. **Use the quadratic formula**:
\[ Q = \frac{-b \pm \sqrt{D}}{2a} = \frac{-16 \pm \sqrt{16}}{2 \times 15} = \frac{-16 \pm 4}{30} \]
5. **Calculate the two roots**:
- First root:
\[ Q_1 = \frac{-16 + 4}{30} = \frac{-12}{30} = -\frac{2}{5} \]
- Second root:
\[ Q_2 = \frac{-16 - 4}{30} = \frac{-20}{30} = -\frac{2}{3} \]

### Step 3: Compare the roots of P and Q

We have:
- Roots of P: \(P_1 = -\frac{3}{2} \) and \(P_2 = -3\)
- Roots of Q: \(Q_1 = -\frac{2}{5} \) and \(Q_2 = -\frac{2}{3}\)

### Step 4: Determine the relationship between P and Q

1. **Convert the roots to decimal for easier comparison**:
- \(P_1 = -1.5\), \(P_2 = -3\)
- \(Q_1 = -0.4\), \(Q_2 = -0.6667\)

2. **Compare the values**:
- \(P_1 = -1.5 < Q_1 = -0.4\)
- \(P_2 = -3 < Q_2 = -0.6667\)

### Conclusion

From the comparisons, we can conclude that both roots of P are less than both roots of Q.

### Final Answer:
**P < Q**

---