What is the simplified quotient? 0 0 (1) \( \frac{x^{2}-6 x+9}{4 x^{2}+20 x+25} \) \( \frac{4 x^{2}+20 x+25}{x^{2}-6 x+9} \)

To simplify the quotients, we first need to factor the polynomials in both the numerator and the denominator. The expression \(x^{2}-6x+9\) can be factored as \((x-3)^{2}\), while \(4x^{2}+20x+25\) can be factored as \((2x+5)^{2}\). Thus, we have: \[ \frac{(x-3)^{2}}{(2x+5)^{2}} \quad \text{and} \quad \frac{(2x+5)^{2}}{(x-3)^{2}} \] The simplified form of each is straightforward—just take the square roots of the fractions: 1. \(\frac{x-3}{2x+5}\) 2. \(\frac{2x+5}{x-3}\) Each gives a simple relationship, one being the reciprocal of the other! Now, watch out for common mistakes! One common pitfall when simplifying expressions like these is forgetting to factor correctly or to simplify all the way. Make sure to check that all terms are factored and reduce! It's like cleaning up—not always easy, but totally necessary for clarity!