The correct answer is:
[C]: ;
x ≠ 5, x ≠ -3 .
___________________________________________Explanation:__________________We are given the following expression:
__________________ ;
__________________ We can factor the "denominator" as follows:
__________________ The denominator: " x² − 2x − 15 " ; can be factored into:
→ " (x−5)(x+3) " ;
→To solve this problem, we want to see if we can factor the number into one of the factors of the denominator; that way we can ultimately "cancel out" one of the same "factors" in both the numerator AND the denominator; (since "any value", divided by that "same value", equals "1". The purpose is to see if we can further simplify the expression (ultimately).
____________ → In our case, we can do so;
→ We can factor out the "numerator" — as follows:
→ The numerator: " 3x² + 9x " ; can be "factored out", into: "(3x)(x+3)" ;
→ Now, we have:"(x+3)" in BOTH the numerator AND the denominator).
→ So, we can rewrite the expression as:
_________________________ ;
_____________At this point, examine the denominator:
→ "(
x−5)(x+3)" .
_____________ Note: The "denominator" cannot equal "0" ; since we cannot "divide by zero".
_____________ So the restrictions on the value for "x" are:
{"any values for "x" in the denominator; that would make the denominator equal to "0" (zero)".
}.
___________ Note that any value, when multiplied by "0" (zero), will result in a product [value] of "0" (zero).
___________ So; we can see that in the [factored] denominator, which is:________________________________________ "(x−5)(x+3)" ; that there are "2 (TWO) multiplicands" , which are:________________________________________________________ 1) "(x−5)" ; AND: 2) "(x+3)" ;