factor
((5a^2+12a+4)/(a^4-16)) / ((25a^2+20a+4)/(a^2-2a))
To simplify the given expression, we need to factor the numerator and the denominator separately.
Let's start with the numerator: (5a^2 + 12a + 4)
We can see that this is a quadratic trinomial, so we need to find two binomials that multiply to give us this trinomial.
To factorize it, we look for two numbers that multiply to give us the product of the coefficient of the quadratic term (a^2 term) and the constant term, and add up to give us the coefficient of the linear term.
In this case, the coefficient of the quadratic term is 5 and the constant term is 4. We need to find two numbers that multiply to give us 5*4 = 20 and add up to give us 12.
The numbers that satisfy this condition are 10 and 2. So we can write the numerator as:
(5a^2 + 10a + 2a + 4)
Now, we can group the terms and factor by grouping:
(5a^2 + 10a) + (2a + 4)
Taking out the common factors, we get:
5a(a + 2) + 2(a + 2)
Notice that we have a common binomial factor of (a + 2). We can now factor it out:
(a + 2)(5a + 2)
Now let's move on to the denominator: (a^4 - 16)
This is a difference of squares, which means we can write it as the product of two binomials:
(a^2 - 4)(a^2 + 4)
Again, we have a difference of squares in the first term, so we can further factorize it:
(a - 2)(a + 2)(a^2 + 4)
Now let's factorize the denominator of the second fraction: (25a^2 + 20a + 4)
We can see that this is a perfect square trinomial. It can be written as the square of a binomial:
(5a + 2)^2
Finally, let's factorize the denominator of the second fraction: (a^2 - 2a)
We can factor out a common factor of 'a' from both terms:
a(a - 2)
Now, putting it all together, we have:
[((5a^2 + 10a + 2a + 4) / (a^2 - 4)(a^2 + 4))] / [(25a^2 + 20a + 4) / a(a - 2)]
Now that we have factored the expression, we are ready to simplify it further or perform any necessary operations.
Factor each of the four terms in parentheses and see which factors you can cancel out. This is not hard. Surely you can factor
a^4 - 16 and a^2 -2a
5a^2 + 12a +4 = (5a +2)(a + 2)
25a^2 +20a +4 = (5a +2)(5a + 2)
Now put it all together