Question

Can someone check my work please? I tried my best on all the problems but I still don't understand excluded values and how to simplify rational expressions. Thank you in advance for any help!
1)6x^5/2x^3
6/2 = 3
3x^5/x^3
x^5-x^3 = x^2
3x^2/1

2)
(2x+4)+(x^2 +2x)
2x+4: 2(x+2)
x^2 +2x: x(x+2)
x(x+2)+ 2(x+2)
(x+2)(x+2)
(x+2)^2
(x+2)^2/x+2
x+2

3)x^3 + 10x^2/ x^2+6x-40
x^2(x+10)/x^2+6x-40
x^2+6x-40: (x-4)(x+10)
x^2(x+10)/ (x-4)(x+10)
x^2/x-4

Answer 1

When simplifying, each new line should have an equal sign at the front
You are saying:
"my next line is equal to the previous line"
Don't just make piece-meal statements

Here is what I would have expected from my students when I was still teaching:

6x^5/(2x^3) ------ those brackets are essential
= 3x^5/x^3
= 3x^2 , x ≠ 0

I would have accepted just the one line solution.
To reach the last line, we divided by x^3.
But if x = 0 , I just divided by zero, which is a mathematical "no-no"
That is why I included the restriction, x ≠ 0

2. (2x+4)+(x^2 +2x)
= 2x + 4 + x^2 + 2x
= x^2 + 4x + 4
or (x+2)^2

I have no idea what all your lines are supposed to be in relation to the problem.
What is the : supposed to mean here. We use the : symbol to express ratios.
How did the addition turn into a ratio ????

3.
I am sure you meant
(x^3 + 10x^2)/(x^2+6x-40) --- again, those brackets are absolutely essential or the only division we would have is 10x^2/x^2

= x^2(x+10)/( (x+10)(x-4) )
= x^2/(x-4) , x ≠ -10

I divided by x+10, so we must restrict the x to x≠-10 or else we divided by zero when we divided by x+10

With the proper use of brackets you would have had the correct answer but without the necessary restriction.