Final answer:
To multiply the given rational expressions, first factor common terms in the numerators and denominators, then multiply and cancel common factors to reach the simplified expression: (75a³(a + 5)) / ((a - 4) * (4a - 1)).
Step-by-step explanation:
To multiply the rational expressions in factored form, we will follow the guidelines by multiplying the numerators and denominators together and then simplifying by eliminating common factors where possible.
The multiplication of the given rational expressions is:
((15a² * 5a) / (7a - 28)) * ((7a + 35) / (20a² - 5a))
Firstly, factor out any common factors in the numerators and denominators:
15a² * 5a = (3 * 5 * a * a * 5a)
7a - 28 = 7(a - 4)
7a + 35 = 7(a + 5)
20a² - 5a = 5a(4a - 1)
Multiplying the numerators and denominators:
((3 * 5 * a * a * 5a) * (7(a + 5))) / ((7(a - 4)) * (5a(4a - 1)))
Now we simplify by canceling out the common factors:
The factor 7 in the numerator cancels with the factor 7 in the denominator. The factor 5a in the numerator cancels with 5a in the denominator.
The simplified form of the expression is:
(3 * 5 * a * a * a + 5) / ((a - 4) * (4a - 1))
Which can also be written as:
(75a³(a + 5)) / ((a - 4) * (4a - 1))
Finally, to ensure the simplified expression is correct, we check if the expression looks reasonable and that all common factors have been eliminated.