What expression would be equivalent to x^2+10x+25/12x/x+5/2x, when x cannot equal 0?
It is not easy to tell.... but I THINK your question is actually...
x^2 + 10x +25
-------------------
12x
_____________________
x+5
------
2x
The first step is to remember the Keep-Change-Flip rule about dividing fractions...
so your problem becomes...
x^2 + 10x +25 2x
------------------- * ------
12x x+5
We can now cancel the x in the 12x denominator with the x in the 2x numerator, and also divide both of these by 2. This gives us...
x^2 + 10x +25 1
------------------- * ------
6 x+5
The next step is to factor x^2 + 10x +25. There are a number of ways to factor this, but it is actually the special case of (x+5)(x+5).... which I usually teach my students to memorize this typical form for...
(x+5)(x+5) 1
------------------- * ------
6 x+5
We can now cancel one of the numerator x+5 terms with the denominator x+5 term
x+5
= -----
6
