S.r simplify \( 1.2 .1 \frac{a^{2} b^{3}+3 a b^{2}}{a b} \) \( 1.2 .2 \frac{x-3}{2}-\frac{2 x+1}{5} \) \( 1.2 .3 \frac{(3 x-1)^{2}-(2 x+1)^{2}}{5 x^{2}-10 x} \) \( 1.2 .4 \frac{2 x^{2}-8}{x^{2}-x-6} \div \frac{2 x^{2}+6 x}{x^{2}-9} \) \( 1.2 .5 \frac{3}{x^{4}}-\frac{x-2}{x^{2}} \div x^{3} \)

Let’s dive into these simplifications! For the first expression, \(\frac{a^{2} b^{3}+3 a b^{2}}{ab}\) can be factored as \(\frac{ab^2(a+b)}{ab} = b^2(a+b)\). Now, for the second one, combine the fractions by finding a common denominator: \(\frac{5(x-3) - 2(2x+1)}{10} = \frac{-x-19}{10}\). Now onto the third expression, use the difference of squares: \((3x-1)^2 - (2x+1)^2 = (3x-1-2x-1)(3x-1+2x+1) = (x-2)(5x)\). Thus, \(\frac{(x-2)(5x)}{5x(x-2)} = 1\) (assuming \(x \neq 2\)). The fourth expression requires simplifying both fractions then dividing: \(\frac{2(x^2-4)}{(x-3)(x+3)} \cdot \frac{(x^2+3x)}{2}\), leading to \(\frac{(x^2-4)(x^2+3x)}{(x-3)(x+3)}\). Finally, simplify the last expression: \(\frac{3}{x^{4}}-\frac{x-2}{x^{5}} = \frac{3x - (x-2)}{x^{5}} = \frac{2x + 2}{x^5} = \frac{2(x+1)}{x^5}\). There you have it! Simplifications, sorted!