Question 912332
The first equation or function is not a rational one, but is a polynomial equation or function.  The function r(x) IS a rational function.  You need to learn what is a polynomial function and then what is a rational function before you can effectively make the sense of these in the way that you are asking.  They ARE related but they are not the same thing.


Moving directly to the r(x), the RATIONAL function,
{{{r(x)=(x(2x+1))/((2-x)(2+x))}}} when factored.


No factor common to both numerator and denominator, meaning no holes.


Undefined for x at 2 and -2, so r(x) has vertical asymptote at x=-2 and at x=2.


x-intercepts for x=0 because it is a factor in the numerator, and for x=-1/2 which makes the binomial 2x+1=0; so x-intercepts  {{{x=-1/2}}} and {{{x=0}}}.


y-intercept:  Let x=0, and evaluate r(0).