Factor completely 9x2 + 42x + 49.

 (3x + 7)(3x − 7) (9x − 7)(9x − 7) (3x + 7)(3x + 7) (9x + 7)(9x + 7)

To factor the polynomial \(9x^2 + 42x + 49\) completely, we will first determine if it is a perfect square trinomial and then factor it accordingly.

### Step-by-Step Solution:

1. **Identify the coefficients:**
   - \(A = 9\), \(B = 42\), \(C = 49\).

2. **Check if the first term is a perfect square:**
   - The first term is \(9x^2\), which is a perfect square since \(9x^2 = (3x)^2\).

3. **Check if the last term is a perfect square:**
   - The last term is \(49\), which is a perfect square since \(49 = 7^2\).

4. **Check the middle term:**
   - The middle term is \(42x\). We need to verify if \(42x = 2 \cdot (3x) \cdot 7\). Indeed, \(42x = 2 \cdot 3x \cdot 7\).

Since all conditions are satisfied, \(9x^2 + 42x + 49\) is a perfect square trinomial.

### Factoring the trinomial:
We can write the trinomial as:

\[9x^2 + 42x + 49 = (3x + 7)^2\]

Thus, the complete factorization of \(9x^2 + 42x + 49\) is:

\[(3x + 7)(3x + 7)\]

So the correct option is:

\[(3x + 7)(3x + 7)\]

Would you like more details or have any questions on this topic?

Here are 8 related questions you might find interesting:
1. How do you factor a trinomial that is not a perfect square?
2. What is the difference between a perfect square trinomial and a difference of squares?
3. How do you solve a quadratic equation using factoring?
4. Can you provide examples of non-factorable trinomials?
5. How do you use the quadratic formula to solve quadratic equations?
6. What are the applications of perfect square trinomials in geometry?
7. How do you factor trinomials with coefficients other than 1?
8. What are the common mistakes to avoid when factoring trinomials?

**Tip:** When factoring trinomials, always check if it can be written as a perfect square trinomial first. This simplifies the factoring process significantly.

Learn how to factor completely the polynomial 9x^2 + 42x + 49 with a step-by-step solution. This problem covers key algebraic concepts including perfect square trinomials. Suitable for Grades 9-12 students.

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