Simplify the following:

First, we can bring both fractions under the common denominator (a - 3)(a - 5):
[tex] \dfrac{4}{a-3}\Big(\dfrac{a-5}{a-5}\Big) +\dfrac{9}{a-5}\Big(\dfrac{a-3}{a-3}\Big)\\\\=\dfrac{4(a-5)+9(a-3)}{(a-3)(a-5)} [/tex]
Looking at the numerator, we can distribute and collect like terms:
[tex] 4(a-5)+9(a-3)=4a-20+9a-27=13a-47 [/tex]
and looking at the denominator, we can do the same:
[tex] (a-3)(a-5)=(a-3)a+(a-3)(-5)\\=a^2-3a-5a+15\\=a^2-8a+15 [/tex]
With these simplified expressions, the final fraction becomes
[tex] \dfrac{13a-47}{a^2-8a+15} [/tex]