Final answer:
To simplify the given expression, factor both the numerator and the denominator, cancel out common factors, resulting in a simplified quotient a(a + 7) / (4a² - 5a + 35).
Step-by-step explanation:
To find the quotient and simplify the expression 2a²+14a divided by 8a²-10a+70, we first factor both the numerator and the denominator. The numerator can be factored by taking out the greatest common factor (GCF), which is 2a:
(2a)(a + 7)
Similarly, the denominator can be factored by taking out the GCF, which is 2:
(2)(4a² - 5a + 35)
We can now simplify the expression by canceling out the common factors:
(2a)(a + 7) / (2)(4a² - 5a + 35)
Canceling the common factor of 2:
a(a + 7) / (4a² - 5a + 35)
Note that the quadratic in the denominator does not factor further and does not share any common factors with the numerator. Hence, the simplified quotient is:
a(a + 7) / (4a² - 5a + 35)
When simplifying expressions, remember to eliminate terms wherever possible and check that the answer is reasonable.