Factor completely, then place the answer in the proper location on the grid.

[tex]\[ 49x^2 + 42xy + 9y^2 \][/tex]

(Note: The instructions on how to interact with the digital interface are not part of the question and have been removed for clarity.)



Answer :

Sure, let's go through the steps to factor the polynomial expression completely.

The given polynomial is:
[tex]\[ 49x^2 + 42xy + 9y^2 \][/tex]

### Step-by-Step Solution:

1. Identify the Structure:
Notice that the expression resembles a perfect square trinomial. A perfect square trinomial takes the form:
[tex]\[ (ax + by)^2 = a^2x^2 + 2abxy + b^2y^2 \][/tex]
Our goal is to express the given polynomial in this form.

2. Rewriting Coefficients:
We need to identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex] such that:
- [tex]\(a^2 = 49\)[/tex], hence [tex]\(a = 7\)[/tex]
- [tex]\(b^2 = 9\)[/tex], hence [tex]\(b = 3\)[/tex]
- [tex]\(2ab = 42\)[/tex], let's verify this:
[tex]\[ 2 \cdot 7 \cdot 3 = 42 \][/tex]

So, the polynomial [tex]\(49x^2 + 42xy + 9y^2\)[/tex] can indeed be written as [tex]\((7x + 3y)^2\)[/tex].

3. Final Factored Form:
Combining these findings, we get:
[tex]\[ 49x^2 + 42xy + 9y^2 = (7x + 3y)^2 \][/tex]

### Conclusion:

The completely factored form of the polynomial [tex]\(49x^2 + 42xy + 9y^2\)[/tex] is:
[tex]\[ (7x + 3y)^2 \][/tex]

This is the final answer.