5) \( \frac{4 a-5}{6 a^{2}+30 a}+\frac{a-1}{6 a^{2}+30 a} \)

To simplify the expression: \[ \frac{4a - 5}{6a^{2} + 30a} + \frac{a - 1}{6a^{2} + 30a} \] **Step 1: Combine the Fractions** Both fractions have the same denominator, so you can combine the numerators directly: \[ \frac{4a - 5 + a - 1}{6a^{2} + 30a} = \frac{5a - 6}{6a^{2} + 30a} \] **Step 2: Factor the Denominator** Factor out the greatest common factor (GCF) from the denominator: \[ 6a^{2} + 30a = 6a(a + 5) \] **Step 3: Rewrite the Expression with Factored Denominator** \[ \frac{5a - 6}{6a(a + 5)} \] **Final Simplified Form:** \[ \boxed{\dfrac{5a - 6}{6a(a + 5)}} \] This is the simplified form of the given expression.