a)

Określamy dziedzinę wyrażenia.

$x-3\ne 0\text{i}{x}^2-9\ne 0\text{i}x+1\ne 0$  

$x\ne 3\text{i}{x}^2\ne 9\text{i}x\ne -1$  

$x\ne 3\text{i}x\ne -3\text{i}x\ne 3\text{i}x\ne -1$  

$D=\mathbb{R}\setminus \left\{-3,-1,3\right\}$

Upraszczamy wyrażenie.

$\frac{\frac{x^2+2x+1}{x-3}}{\frac{x+1}{x^2-9}}=\frac{x^2+2x+1}{x-3}\cdot \frac{x^2-9}{x+1}=$  

$=\frac{{\left(x+1\right)}^2}{x-3}\cdot \frac{\left(x-3\right)\left(x+3\right)}{x+1}=$

$=\left(x+1\right)\left(x+3\right)$

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b)

Określamy dziedzinę wyrażenia.

$x^2+4x+4\ne 0\text{i}{x}^2-4\ne 0\text{i}4x^2-4x+1\ne 0$  

${\left(x+2\right)}^2\ne 0\text{i}{x}^2\ne 4\text{i}{\left(2x-1\right)}^2\ne 0$  

$x+2\ne 0\text{i}x\ne 2\text{i}x\ne -2\text{i}2x-1\ne 0$  

$x\ne -2\text{i}x\ne 2\text{i}2x\ne 1\mid :2$  

$x\ne -2\text{i}x\ne 2\text{i}x\ne \frac{1}{2}$  

$D=\mathbb{R}\setminus \left\{-2,\frac{1}{2},2\right\}$

Upraszczamy wyrażenie.

$\frac{\frac{4x^2-1}{x^2+4x+4}}{\frac{4x^2-4x+1}{x^2-4}}=\frac{4x^2-1}{x^2+4x+4}\cdot \frac{x^2-4}{4x^2-4x+1}=$

$=\frac{\left(2x-1\right)\left(2x+1\right)}{{\left(x+2\right)}^2}\cdot \frac{\left(x-2\right)\left(x+2\right)}{{\left(2x-1\right)}^2}=$

$=\frac{2x+1}{x+2}\cdot \frac{x-2}{2x-1}=\frac{\left(2x+1\right)\left(x-2\right)}{\left(x+2\right)\left(2x-1\right)}$