x^2+8x+7<0...................basic stuff but intriguining.

Welcome to GMAT club.

The rule is once u get the solution you need to ascertain its validity or you need to plug in that solution back in equation and test for validity. for example:

for first equation, once you've reached to (x+1)(x+7)<0
Now you know that for this to be true, one term must be negative and other positive.
say (x+1) is negative => x <-1
but at this time x+7 must be positive, so x+7 > 0 => x>-7
Hence we get the range -7 <x<-1

Again second possibility,
if x+1 is positive then x+7 should be negative or
x+1 >0 => x >-1
and x+7 <0 => x < -7
but it is not possible. Since x can not be greater than -1 and less than -7 at same time.

So you get that the solution obtained first was correct one. and range is -7<x<-1
similarly you can solve the second problem.

Once you are comfortble getting ans like this, then slowly u can move to graphical approach which makes it even easier to get the answer by knowing the concept.

Hope it helps.