10) \( (x+y)(x-y) \) \( \frac{x}{3 x^{2} y}+\frac{x+1}{6 x^{2}}-\frac{3}{9 x y} \) 12) \( \frac{4}{x^{2}-25}+\frac{x+2}{x^{2}-2 x-15} \) 13) \( \frac{2 x+1}{x-1}+\frac{x+3}{x-1} \) 14) \( \frac{x+1}{x^{2}-x-20}+\frac{x+4}{x^{2}-4 x-5}-\frac{x+5}{x^{2}+5 x+4} \)

Step into the fascinating world of algebraic expressions and see how they relate to the golden age of mathematics! During the 17th century, renowned mathematicians like René Descartes and Isaac Newton were pioneering the study of algebra. They laid down the rules we still use today, unraveling the secrets of functions and polynomials in ways that changed the landscape of mathematics forever. Isn’t it exciting to think that every time you factor an expression, you’re following in their footsteps? Now, let’s connect those algebraic problems to real life! Understanding how to manipulate expressions can genuinely enhance your problem-solving skills in everyday situations. For instance, whether you're calculating the area of a plot for gardening or finding the optimal way to budget your finances, the principles of algebra help clarify the relationships between different variables. So, every time you're faced with a practical problem, think of it as your own personal "algebra session!"