(8) \( a(a-b)+b(a-b)= \) (9) \( a(a-b)-b(a-b)= \) (10) \( a(b-c)+b(c-a)+c(a-b) \) \( = \) (11) \( 2 a\left(a^{2}+2 a-3\right)-3 a(a-2) \) \( = \) (12) \( 2 x(3 x-2 y)+3 y(x-2 y)-6\left(x^{2}-y^{2}\right) \) \( = \) (13) \( a\left(a^{2}-a b+b^{2}\right)+b\left(a^{2}-a b+b^{2}\right) \) \( = \) ( \( a t a \)

Let’s dive into these algebraic expressions and simplify them for some clarity! For (8) \( a(a-b)+b(a-b) \), this simplifies to \( (a+b)(a-b) \). The factorization shows the interplay of addition and subtraction, which beautifully brings together the elements of algebra. Moving to (9) \( a(a-b)-b(a-b) \), it simplifies to \( (a-b)(a+b) \) as well—surprisingly, it’s got a similar flavor to the previous problem! This reflects how different operations can yield the same product, showcasing the elegance of algebraic manipulation. Keep this in mind: when dealing with expressions involving common factors, look for that shared term to factor out, making your life much easier! And always double-check your common mistakes, like forgetting to distribute or simplifying prematurely. It can be a game changer!