Question

For what values of `x` is the relation `((-2x-10)(x-3))/((x^2+2)(x-2)^2) < 0` true?

Answer 1

The inequality is true for `x in (-5,+3)`
BUT your real interest is in understanding how to get this... see below.

Explanation:

Given
`color(white)("XXX")((-2x-10)(x-3))/((x^2+2)(x-2)^2) < 0`

Assuming `x!=2` (in which case the left side is undefined)
we note that both `(x^2+2)` and `(x-2)^2` must be non-negative values.

So we can simplify the given inequality by multiplying both sides by `(x^2+2)(x-2)^2`
to get
`color(white)("XXX")(-2x-10)(x-3) < 0`

There are (only) two critical values for `x` which would make the left side equal to zero,
namely `x=-5` and `x=3`

These two values divide the Domain into 3 pieces (plus the two critical points):
#{: (," | ","values"," | ",," | ","values between"," | ",," | ","values"), (x=," | ",< -5," | ", -5," | ",(-5,+3)," | ",+3," | ", >+3), ((-2x-10)(x-3)=," | ",?," | ",0," | ",?," | ",0," | ",?) :}#

We can pick any value(s) we like in the non-specific ranges and test to see if the desired inequality holds:
#{: (," | ","values"," | ",," | ","values between"," | ",," | ","values"), (x=," | ",< -5," | ", -5," | ",(-5,+3)," | ",+3," | ", >+3), ("sample value:"," | ",-6," | ",," | ",0," | ",," | ",4), ((-2x-10)(x-3)=," | ",18," | ",0," | ",-30," | ",0," | ",18), ("is this " < 0," | ","No"," | ","No"," | ","Yes"," | ","No"," | ","No") :}#

From this we see that the given inequality is only true for
`color(white)("XXX") x in (-5,+3)`