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.
