Final answer:
To sum the two given rational expressions, we find a common denominator, which is (25-9z²), and then add the adjusted numerators to get the final simplified expression (-6z - 10) / (25 - 9z²).
Step-by-step explanation:
The student is asking to find the sum of two rational expressions: (4)/(3z+5) and (6z-30)/(25-9z²). To add these, a common denominator must be found. The second denominator factors as (25 - 9z²) = (5-3z)(5+3z). Since 3z+5 is already part of this, the first fraction can be rewritten over this common denominator by multiplying the numerator and denominator by (5-3z). Doing so, we have:
(4)(5-3z)/(25 - 9z²) + (6z-30)/(25-9z²)
The numerators can now be added:
(20 - 12z + 6z - 30) / (25 - 9z²) = (-12z + 6z + 20 - 30) / (25 - 9z²)
Simplifying the numerator yields:
(-6z - 10) / (25 - 9z²)
Which is the simplified form of the given expression.