5x/x^2+3x+2 minus 2x-6/x^2+4x+4?

Brackets required.
Assume you mean, NOT as given, but as :-

5x / (x² + 3x + 2) - (2x - 6) / (x² + 4x + 4)

5x / (x + 2)(x + 1) - (2x - 6) / (x + 2)²

5x (x + 2) - (2x - 6) (x + 1)
----------------------------------
(x + 2)² (x + 1)

5x² + 10x - 2x² + 4x + 6
------------------------------
(x + 2)² (x + 1)

3x² + 14x + 6
--------------------------
(x + 2)² (x + 1)

5x / x^2 +3x + 2 minus 2x - 6/ x^2 + 4x + 4

1) You can perform this operation only if the denominator is the same

2) factor the denominator and/ or the numerator to simplify the terms

5x / ( x + 2) ( x + 1) minus 2x - 6/( x + 2) ( x + 2) or (x + 2)^2)

3) Our common denominator is

(x + 1) ( x + 2)^2

Since (x + 2) is missing in the denominator of the first term, we need to multiply it to the numerator. Since ( x + 1) is missing in the denominator of the second term, we need to mulitply it to the numerator of the second term:

5x( x+ 2) / ( x + 1) ( x+ 2)^2 minus (x + 1) (2x - 6)/( x+ 1) ( x + 2)^2 =

5x^2 + 10x/(x+1) (x +2)^2 minus 2x^2 - 4x -6/(x+1)(x+2)^2 =
Remember - (2x^2 - 4x - 6) becomes -2x^2 + 4x + 6

(3x^2 + 14x + 6 )/ (x +1) (x +2)^2 ==> the answer

x^2 +3x +2 = (x+1)*(x+2)
x^2+4x+4 = (x+2)^2
the given function = [5x/(x+1)*(x+2)] - [ 2x -6/(x+2)*(x+2)]
If in the problem the second function is (2x-6)/ x^2+4x+4, the answer will be different from what i do below.
The common denominator will be (x+1)*(x+2)*(x+2)
The given fn. = {5x(x+2) - 2x(x+1)(x+2)(x+2) + 6(x+1)}/ (x+1)(x+2)(x+2)
=[5x^2 + 10x - 2x^4 - 10x^3 - 16x^2 - 8x +6x+6] / (x^3+5x^2+8x+4)
=[6 +8x - 11x^2 - 10x^3 -2x^4] / (x^3+5x^2+8x+4)
I am also a mother and i can understand your concern.

5x/(x^2 + 3x + 2) - (2x - 6)/(x^2 + 4x + 4)
= 5x/(x^2 + 2x + x + 2) - (2x - 6)/(x^2 + 2x + 2x + 4)
= 5x/(x + 2)(x + 1) - (2x - 6)/(x + 2)(x + 2)
= 5x(x + 2)/(x + 2)^2(x + 1) - (2x - 6)(x + 1)/(x + 2)^2(x + 1)
= (5x*x + 5x*2)/(x + 2)^2(x + 1) - (2x*x - 6*x + 2x*1 - 6*1)/(x + 2)^2(x + 1)
= (5x^2 + 10x)/(x + 2)^2(x + 1) - (2x^2 - 4x - 6)/(x + 2)^2(x + 1)
= (5x^2 + 10x - 2x^2 + 4x + 6)/(x + 2)^2(x + 1)
= (3x^2 + 14x + 6)/(x + 2)^2(x + 1)