Let's go step by step to factorize each of these expressions:

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i. $6x(2x - y) + 7y(2x - y)$

Here, both terms share the common factor $(2x - y)$.

$6x(2x - y) + 7y(2x - y) = (2x - y)(6x + 7y)$

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ii. $2r(y - x) + s(x - y)$

Notice that $(x - y)$ is the negative of $(y - x)$, so we can factor out $(y - x)$ as a common factor.

$2r(y - x) + s(x - y) = 2r(y - x) - s(y - x) = (y - x)(2r - s)$

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iii. $9a(6a - 5b) - 12a^2(6a - 5b)$

Here, we see that both terms share the common factor $(6a - 5b)$, so we factor it out.

$9a(6a - 5b) - 12a^2(6a - 5b) = (6a - 5b)(9a - 12a^2)$

Factor the second term further:

$(6a - 5b)(3a(3 - 4a))$

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iv. $(x - y)^2 + (x - y)$

We can factor out the common factor $(x - y)$:

$(x - y)^2 + (x - y) = (x - y)((x - y) + 1) = (x - y)(x - y + 1)$

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v. $a(x - y) + 2b(y - x) + c(x - y)^2$

We notice that $(y - x) = -(x - y)$, so we can rewrite the terms and factor them out.

$a(x - y) + 2b(y - x) + c(x - y)^2 = a(x - y) - 2b(x - y) + c(x - y)^2$

Factor out $(x - y)$:

$= (x - y)(a - 2b + c(x - y))$

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vi. $x^3(a - 2b) + x^2(a - 2b)$

Factor out the common factor $x^2(a - 2b)$:

$x^3(a - 2b) + x^2(a - 2b) = x^2(a - 2b)(x + 1)$

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Summary of Factorized Forms:

1. $(2x - y)(6x + 7y)$
2. $(y - x)(2r - s)$
3. $(6a - 5b)(3a(3 - 4a))$
4. $(x - y)(x - y + 1)$
5. $(x - y)(a - 2b + c(x - y))$
6. $x^2(a - 2b)(x + 1)$

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Would you like further details on any of these steps? Let me know if anything is unclear!

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Here are some related questions to deepen understanding:

1. How do you factor expressions with common terms and signs like in question ii?
2. Can you explain how to handle factoring when terms have powers, like in question vi?
3. What happens when the factor is squared, as seen in question iv?
4. How do you deal with negative factors in factorization?
5. What is the strategy when dealing with binomial expressions in factorization?

Tip: Always look for common terms first, as they can simplify your work significantly!