Ineed with number 41. find the value of x. round to the nearest hundredth, if necessary.

 x = 4.87 .

   A = L * w  ;

(Area = Length * width ) 

 Given:  Area = 288 in² .
           
            Length = (2x + 10) in
            
            Width =  (3x)  in 

         A = L * w   ;

  288 =  (2x + 10) * (3x) ; 

Solve for "x" ;

  Write as:

(3x)*(2x + 10) = 288 ;

  Note the distributive property of multiplication:

  a(b+c) = ab + ac ; and:
  
  a(b−c) =  ab − ac ; 

So; 

  (3x)*(2x + 10) =
  
   (3x)*(2x)  +  (3x)*(10)  =
  
    6x²  +  30x = 288 ;

Subtract "288" from each side of the equation:

    6x²  +  30x − 288 = 288 − 288 ;

    6x²  +  30x − 288 = 0 ;

 Now, divide the ENTIRE equation by "6";

    {6x²  +  30x − 288} / 6  = 0 / 6 ;

      x²  +  5x − 48 = 0 ; 

We can solve for "x" using the quadratic formula; 

 Note, this equation:

      x²  +  5x − 48 = 0 ;  is written is "quadratic format"; that is:

  ax² + bx + c = 0 ; a ≠ 0 ;

in which:
             a = 1 (the implied coefficient of "1", since anything multiplied by                              "1" is that same value);
              b = 5 ; 
              c = -48 ;

and we can solve for "x" using the quadratic equation formula; which is:
 
             x =  { -b ± √(b² − 4ac) } / {2a} ;  a ≠ 0 ;
            
So, we can plug in the values: 

        First, let's simplify: "(b² − 4ac)" ; which is: 

                        5² − 4*1*(-48) = 25 − (-192) = 25 + 192 = 217 ;

 So, x = -5  ± √(217) / (2*1) ;

       x = { -5  ± √(217) } / 2 ;

So, there are 2 (TWO) values:

Start with: 

 x =  { -5 + √(217) } / 2  = 4.8654599313281177

then, { -5 − √(217) } / 2 = -19.7309198626562354

Since we are dealing with a rectangle, the answer is the positive value; that is:  4.8654599313281177 ;  the question asks to round to the nearest hundredth, if necessary; so;  →  4.87 = x .