The expression you've given is a quadratic expression:

\[
49x^2 - 14x + 1
\]

To factor this quadratic expression, we can check if it's a perfect square trinomial, which has the form \((ax + b)^2 = a^2x^2 + 2abx + b^2\). Let's see if it fits this pattern:

1. **Identify coefficients:**
   - \(a^2 = 49\), so \(a = 7\) because \(7^2 = 49\).
   - \(c = 1\), so \(b = 1\) because \(1^2 = 1\).

2. **Check the middle term:** The middle term should be \(2ab\):
   - \(2 \cdot 7 \cdot 1 = 14\), and indeed the middle term of the given expression is \(-14x\), which matches if we consider it with a negative sign.

Thus, the given quadratic can be rewritten as:

\[
49x^2 - 14x + 1 = (7x - 1)^2
\]

This is a perfect square trinomial.

### Final Answer
\[
49x^2 - 14x + 1 = (7x - 1)^2
\]

Would you like more details or have any questions?

Here are some related questions that could expand on this topic:

1. How do you recognize a perfect square trinomial in general?
2. What if the quadratic was \(49x^2 + 14x + 1\)? Would it still be a perfect square trinomial?
3. How do you solve the quadratic equation \(49x^2 - 14x + 1 = 0\) using factoring?
4. Can a non-perfect square trinomial be factored, and how would you do it?
5. What methods can be used to factor trinomials that are not perfect squares?

**Tip:** When factoring quadratics, always check for a perfect square trinomial first, as it can significantly simplify the process.

Learn how to factor the quadratic expression 49x^2 - 14x + 1 using the perfect square trinomial method. This problem demonstrates the application of factoring perfect square trinomials, suitable for high school students in Grades 9-12.

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