How do you simplify (a/a+2a+1)+(3/4a^2-4)?

Question

I've simplified it down to (a/3a+1)+(3/4(a^2-1)) but I don't know if I did right first step because I'm not sure where to go next. 

Answer 1

 

Chani,  Given this description

 

   a                 3
-------- + ----------
(3a+1)      4(a^2-1)

 

We first need a common denominator

 

The left fraction has (3a+1) as its only factor, the right fraction has 4(a+1)(a-1) as its factors

 

So the common denominator is 4(a+1)(a-1)(3a+1) [Don't multiply it out yet, it might simplify later].

 

The numerator of the left fraction has to be multiplied by the factors in the denominator of the right fraction and becomes  4(a)(a+1)(a-1) = 4a (a^2 -1) = 4a^3-4a

 

The numerator of the right fraction needs to be multiplied by the factors in the denominator of the left fraction and becomes 3(3a+1) = 9a+3

 

Now we add the numerators and get 4a^3 + 5a + 3 which factors as (2a+1)*(2a^2-a+3).

 

So the final answer does not simplify and in factored terms is

(2a+1)(2a^2-a+3)

---------------------

4(a+1)(a-1)(3a+1)

 

or you can multiply it out.  Check again that you have the fractions right.  It's slightly unusual that something doesn't cancel out in problems like this.  A switched sign (i.e. minus for plus or plus for minus) might result in something cancelling out.