1. **State the problem:** Factor the perfect square trinomial $$289x^2 + 34x + 1$$. 2. **Recall the formula for a perfect square trinomial:** $$a^2 + 2ab + b^2 = (a + b)^2$$ This means the trinomial can be factored into the square of a binomial if it fits this pattern. 3. **Identify terms:** - The first term is $$289x^2$$ which is $$(17x)^2$$. - The last term is $$1$$ which is $$1^2$$. 4. **Check the middle term:** The middle term should be $$2ab = 2 \times 17x \times 1 = 34x$$, which matches the given middle term. 5. **Write the factorization:** Since the trinomial fits the perfect square pattern, it factors as: $$289x^2 + 34x + 1 = (17x + 1)^2$$. 6. **Conclusion:** The polynomial is not prime; it factors completely as $$(17x + 1)^2$$.