Determine the neutral element of the law of composition x*y = 3xy + 6(x+y) + 10

To determine the neutral element we'll write it's
property:

x*e = e*x = x

Now,
we'll apply the law of composition for

 x*e = 3xe + 6(x+e)
+ 10

But x*e = x

3xe + 6(x+e)
+ 10 = x

We'll remove the
brackets:

 3xe + 6x + 6e + 10 =
x

We'll isolate all the elements that contain e to the left
side:

3xe + 6e = x - 6x -
10

We'll factorize by 3e to the left side and we'll combine
like terms to the right side:

3e(x + 2) = -5x -
10

We'll factorize by -5 to the right
side:

3e(x + 2) =
-5(x+2)

We'll divide by (x+2) and we'll
get:

3e = -5

The
neutral element is e = -5/3.