(x+ x/y) *( x- x/y)

First we will simplift the expression:

E = (x + x/y) * (x- x/y)

= ( xy+ x)/y ]* [(xy - x)/y]

= (xy+x)(xy-x)/y^2

We know that: (a+ b)(a-b) = a^2 - b^2

==> E = [ (xy)^2 - x^2]/y^2

Now given x= 30 y= 15

==> E = (30*15)^2 - 30^2 ]/ (15)^2

= (450)^2 - 900]/(225)

= 896

We notice that the product could be transformed into a difference of squares:

(x+x/y)*(x-x/y) = x^2 - (x/y)^2

x^2 - (x/y)^2 = x^2 - x^2/y^2

We'll factorize by x^2:

x^2 - x^2/y^2 = x^2*(1 - 1/y^2)

Now, we'll substitute x and y into the given expression:

x^2*(1 - 1/y^2) = 900*(1 - 1/225)

We'll re-write the expression:

900*(1 - 1/225) = 900(225-1)/225

900*(1 - 1/225) = 900*224/225

900*(1 - 1/225) = 4*224

900*(1 - 1/225) = 896

We also could write the difference of squares, 25 - 1, as:

225 - 1 = (15-1)(15+1) = 14*16

**(x+x/y)*(x-x/y) = 896**

To find the product (x+x/y)((x-x/y) for x= 30 and y = 15.

Let us first simplify the expression and then put x= 30 and y= 15.

(x+x/y)(x-x-x/y ) = x^2-(x/y)^2 , as (a+b)(a-b) = a^2-b^2.

Now we pur x= 30 and y = 15 in x^2-(x/y)^2. Then the value of the expression is 30^2 -(30/15)^2 = 900- 2^2 = 900-4 = 896.

Now we put x = 30 and y = 15 in the directly in the expression (x+x/y)(x-x/y) = (30+30/2)(30-30/2) = 32*28 = 996. We can go by one of the method which is easier to the context getting the same result.