Which of the binomials below is a factor of this trinomial?
5х2 - 5x - 100
а. х-4
ов. x+ 10
ос. x+5
od. х- 5

 The correct answer is:   [D]:  " (x - 5) "

Step-by-step explanation:

First, let's try to "factor out" the given trinomial:

   "  5x² -  5x  - 100 " ;

We can "factor out" a "5" ;

since among the three (3) terms in this trinomial,

 We have:  5x² , 5x, and 100 ;

Note that not all of these 3 terms have a variable in them;

           → Consider the value;  "100" .

Note that the remaining two (2) terms have the same variable, "x" ;  

          →  5x² ; and 5x " ;

Note that each of these 2 (two) terms have a coefficient of "5" ;

Note that the other term, "100" ; is divisible "evenly"  by "5" ;

         → that is; when the term, "100", is divided by "5" ; the result is an integer;  {specially, " 100 ÷ 5 = 20 " .}.

So:  Let us being by "factoring out" a "5" ;

Given the trinomial:  " 5x² - 5x - 100 " ;

         →5(x² - 1x - 20) ;  or:  5(x² - x - 20) ;

       → At this point, although this trinomial express is not completely

"factored out" ;

Let us consider:  Do either of these two (2) factors appear among the answer choices given:

          1)  " 5" ;  or:  2)   "(x² - x - 20)" ??  No!

Let us continue.

We have:  5(x² - x - 20) ;  

       →  We can write as:  " 5 (x² - 1x - 20) "  ;

Now, let see if we can further "factor out" the:

       →  (x² - 1x - 20) ;

Now, note the " -20" ; and the " -1x " ;

      → What two (2) factors of:  "-20" ;  add up to "-1" ?  ;

Let us consider the factor of "-20" :

-20,  1 :  -20 + 1  = -19 ;  -19 =? -1 ?  No.

 20, -1 :  20 + (-1) = 20 - 1 = 19 ;  19 =? -1 ? No.

 10, -2 :  10  + (-2) = 10 - 2 = 8 ; 8  =? -1 ?  No.

-10,  2 :  -10 + 2 =  -8 ;  -8  =? -1 ?  No.

 -4, 5  :  - 4 + 5 = 5 + (-4) = 5 - 4 = 1 ;  1  =? -1 ?  No.

 4, -5  :   4 + (-5) = -5 + 4 = -1 ;  -1 =? -1 ?  Yes!

So,  we have +4  and -5 ;  

  (+4) * (-5) = -20 ;

  (+4) + (-5)  =  -5 + (+4) =  -5 + 4 = -1 ;

So, we factor out:

 →  (x² - 1x - 20) ;

that is:  " (x² - x - 20) " ;

into:

" (x + 4) (x - 5) " ;

and to factor out:  " 5x² - 5x - 100 " ;

we bring down the "5" ;  and write as:

 →  " 5 (x + 4) (x - 5) " ,

So:

We have three (3) factors when the trinomial expression is "factored out":  

 →  5 ;  (x + 4) ;  and:  (x - 5) ;

 → Among these three factors, the only one that appears as a given answer choice is:  

 →  " (x - 5) " ;

 →  which is our

 →   Answer choice:  [D]:  " (x - 5) " .

Hope this answer is helpful to you!

      Best wishes!

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