Which function is a quadratic function? p(x)=2x(×^2+6)+1 m(x)=-4(x+3)-2 t(x)=-8x^2(x^2-6+1 h(x)=3x(x-2)-4

The correct answer is:  [D]:  " H(x) = 3x(x - 2) - 4 " .

Note:  A "quadratic function" takes the form of:

     f(x)  =  ax²  + bx  + c ;

Answer choice:  [D]:  " H(x) = 3x(x - 2) - 4 " ; 

takes this form:

    " H(x) = 3x(x - 2) - 4 " ;  

→  " 3x(x - 2) - 4 " ; 

Note the "distributive property" of multiplication:


  a(b + c)  =  ab  +  ac ;  AND

  a(b -  c)  =  ab  -   ac ;

 
 → As such;

    " 3x(x - 2)  =  (3x * x)  -  (3x * 2)  = 3x² - 6x;

So, we can rewrite the expression:

→  " 3x(x - 2) - 4 " ;  

as:  →  " 3x² - 6x - 4 ; 

And the entire function as:

H(x) = 3x(x - 2) - 4 ; 

as:  

H(x) = 3x² - 6x - 4 ; 

Which takes the form of a "quadratic function" ; 

→ f(x) = ax² + bx + c ; 

in which:  a = 3 ;   b = - 6 ;  c = - 4 .