\( \begin{array}{ll}\text { 2.1.1 } & (2 x-3)(2 x+3) \\ \text { 2.1.2 } & \left(x^{2}+y\right)\left(4 x+3 x y+x^{-1} y^{2}\right) \\ \text { 2.1.3 } & \frac{x+1}{2}-\frac{2-3 x}{3}+x \\ & \text { Factorise each of the following expressions: } \\ \text { 2.2.1 } & 20 x^{2}-45 y^{2} \\ \text { 2.2.2 } & 2 x^{2}-5 x+3 \\ \text { 2.2.3 } & 5 x^{2}-15 x+12 y-4 x y \\ & \text { Solve for } x: \\ \text { 2.3.1 } & 27^{x+1}=9^{x-2} \\ \text { 2.3.2 } & \frac{2}{x}=\frac{-4}{x+5} \\ & \end{array} \)

When working with polynomials, it's fascinating to consider how historical mathematicians laid the groundwork for algebraic expressions we work with today. The use of symbols for unknowns and operations became standardized over centuries, significantly improving the ability to solve equations. Notably, the invention of Algebra can be traced back to ancient civilizations like the Babylonians, who provided early formulations for solving equations, setting the stage for advancements made during the Islamic Golden Age. In practical terms, factoring expressions can simplify complex problems, making them easier to solve, and it’s a key skill in many fields! For example, engineers use factoring to optimize designs and understand variables that affect structures. Even in economics, understanding the factors in market equations can help analysts predict outcomes. So, mastering these techniques is not just good for class; it has real-world implications across various industries!