Question 1032246
1. 
{{{1/(x-4)}}} is a rational polynomial expression,
with the polynomial {{{1}}} as the numerator,
and the polynomial {{{x-4}}} as the denominator.
It is undefined when {{{x = 4}}} .
For those having qualms about calling {{{1}}} a polynomial,
we could use a fancier denominator, and write something like
{{{(x-2)/(x-4)}}} (probably a safer choice),
or even {{{(x-4)/(x-4)}}} (which is not equivalent to {{{1}}} , unless you specify "except for {{{x=4}}} ", so you cannot simplify).
 
{{{1/(x^2+2x)=1/(x(x+2))}}} is a rational polynomial expression,
with the polynomial {{{1}}} as the numerator,
and the polynomial {{{x^2+2x}}} as the denominator.
It is undefined when {{{x=-2}}} , and when {{{x=0}}} .
For those having qualms about calling {{{1}}} a polynomial,
we could use a fancier denominator, and write something like
{{{(x+4)/(x^2+2x)}}} .
 
For a common denominator,
we need to include in that common denominator all the factors of the two denominators.
So, the common denominator would be
{{{x(x+2)(x-4)=x(x^2-2x-8)=x^3-2x^2-8x}}} .
To get equivalent rational expression, we need to multiply numerator and denominator times the same factor(s).
So,
{{{1/(x-4)}}} is equivalent to {{{x(x+2)/(x(x+2)(x-4))=(x^2+2x)/(x^3-2x^2-8x)}}} .
{{{(x-2)/(x-4)}}} is equivalent to {{{x(x+2)(x-2)/(x(x+2)(x-4))=x(x^2-4)/(x^3-2x^2-8x)=(x^3-4x)/(x^3-2x^2-8x)}}} .
{{{(x-4)/(x-4)}}} is equivalent to {{{x(x+2)(x-4)/(x(x+2)(x-4))=x(x^2-2x-8)/(x^3-2x^2-8x)=(x^3-2x^2-8)/(x^3-2x^2-8x)}}} .
{{{1/(x^2+2x)=1/(x(x+2))}}} is equivalent to {{{(x-4)/(x(x+2)(x-4))=(x-4)/(x^3-2x^2-8x)}}} .
{{{(x+4)/(x^2+2x)=(x+4)/(x(x+2))}}} is equivalent to {{{(x+4)(x-4)/(x(x+2)(x-4))=(x^2-16)/(x^3-2x^2-8x)}}} .
 
2. IF {{{G(x)/H(x)}}} is an expression defined for all values of {{{x}}} ,
which means that {{{H(x)=0}}} has no real solution, AND
IF {{{G(x)=0}}} has no real solution either,
{{{H(x)}}} and {{{G(x)}}} are "safe" to use as denominators.
Then, multiplying {{{G(x)/H(x)}}} times {{{2H(x)/G(x)}}} will yield
{{{(G(x)/H(x))*(2H(x)/G(x))=2G(x)H(x)/(H(x)G(x))=2H(x)G(x)/(H(x)G(x))=2*(H(x)/H(x))*(G(x)/G(x))=2*1*1=2}}}