Simplify the expressions in each pair. (2 marks each) (Show your work)

i) (x^2-x-2)/(x+1),g(x)=(x-2)
My work on simplifying the functions:
(x-1)(x+2)/(1)
(x-1)(x+2)

ii) (x^2-x-12)/(x+3), g(x)=(x^2-6x+8)/(x-2)
My work on simplifying the functions:
(x-2)(x+6)/(3)
(x-4)

Can you someone show me where I went wrong and show me the correct answer if I did go wrong.

(x-2)(x+1)/(x+1) = x-2 = g(x)

(x-4)(x+3)/(x+3) = x-4

g(x) = (x-2)(x-4)/(x-2) = x-4 also

i) Let's simplify the expression (x^2-x-2)/(x+1) using long division:

x - 2
_______________________
x + 1 | x^2 - x - 2
- (x^2 + x)
_______________________
-2x - 2
- (-2x - 2)
_______________________
0

As you can see from the long division, the expression simplifies to x - 2, not (x-1)(x+2).

ii) Let's simplify the expression (x^2-x-12)/(x+3) using long division together with the given function g(x) = (x^2-6x+8)/(x-2):

x - 4
____________________
x + 3 | x^2 - x - 12
- (x^2 + 3x)
____________________
-4x - 12
- (-4x - 12)
____________________
0

Again, from the long division, the expression simplifies to x - 4, not (x-2)(x+6)/(3).

Therefore, the correct simplified expressions are:

i) (x^2-x-2)/(x+1) simplifies to x - 2.

ii) (x^2-x-12)/(x+3) simplifies to x - 4.