1. **State the problem:** Simplify or factor the quadratic expression $x^2 - 34x + 289$. 2. **Recall the formula:** To factor a quadratic expression of the form $ax^2 + bx + c$, we look for two numbers that multiply to $ac$ and add to $b$. 3. **Identify coefficients:** Here, $a=1$, $b=-34$, and $c=289$. 4. **Check if the quadratic is a perfect square:** Since $289 = 17^2$ and $-34 = -2 \times 17$, this suggests the quadratic might be a perfect square. 5. **Write as a perfect square:** $$x^2 - 34x + 289 = (x - 17)^2$$ 6. **Verify by expansion:** $$(x - 17)^2 = x^2 - 2 \times 17 \times x + 17^2 = x^2 - 34x + 289$$ 7. **Conclusion:** The quadratic expression factors perfectly as $(x - 17)^2$. **Final answer:** $$(x - 17)^2$$