a) am-an-2bn+2bm

b) 2ax-6bx+ay-3by

c) 2ax-6bx-ay+3by

d) a^2b+c^2b-c^2b-a^2b

e) a^3+3a^2+2a+6

f) ax-2ay+2bx-4by

I will do a couple , you do the rest following the same method

b)
2ax-6bx+ay-3by
= 2x(a - 3b) + y(a - 3b)
you MUST have the same binomial in each of the terms

= (a - 3b)(2x + y)

or

2ax + ay - 6bx - 3by)
= a(2x + y) - 3b(2x + y)
= (2x+y)(a - 3b)
showing that there is more than one way to do these

f) ax - 2ay + 2bx - 4by
= a(x - 2y) + 2b(x - 2y)
= (x-2y)(a+2b)

To simplify each expression, we can combine like terms by grouping them together. Like terms have the same variables raised to the same powers.

a) am - an + 2bn + 2bm
- This expression contains two different variables, m and n. To simplify, group the terms with the same variable together:
am + 2bm - an + 2bn

b) 2ax - 6bx + ay - 3by
- In this expression, we have two different variables, x and y. Group the terms with the same variable together:
2ax - 6bx + ay - 3by

c) 2ax - 6bx - ay + 3by
- Similar to the previous expression, we group the terms with the same variable together:
2ax - 6bx - ay + 3by

d) a^2b + c^2b - c^2b - a^2b
- In this expression, we have terms with the same variables raised to the same powers. Group them together:
(a^2b - a^2b) + (c^2b - c^2b)
Simplifying further, the terms cancel each other out, so the answer is 0.

e) a^3 + 3a^2 + 2a + 6
- In this expression, we have terms with just the variable 'a', but raised to different powers. There are no like terms to combine here, so the expression remains as it is.

f) ax - 2ay + 2bx - 4by
- In this expression, we have two different variables, x and y. Group the terms with the same variable together:
ax + 2bx - 2ay - 4by

Remember, to simplify expressions by combining like terms, look for terms with the same variables and powers, then combine them by performing the necessary addition or subtraction.