(a) \( A_{3}=\left\{\frac{2 n^{2}-n-10}{2 n^{2}+3 n-2}: n \in \mathbb{Z}\right\} \) (b) \( A_{4}=\{x \in \mathbb{Q}: x \in(0,6]\} \)

**(a) Finding an alternate description for** \[ A_{3}=\left\{\frac{2n^{2}-n-10}{2n^{2}+3n-2}: n\in\mathbb{Z}\right\}. \] 1. **Factor the denominator.** Notice that \[ 2n^2+3n-2 = (2n-1)(n+2). \] 2. **Factor the numerator.** We look at \[ 2n^2 - n -10. \] Compute the discriminant: \[ D = (-1)^2 - 4\cdot2\cdot(-10)= 1+80=81. \] Since \( \sqrt{81}=9 \) the roots are \[ n=\frac{1\pm 9}{4} \quad \Longrightarrow \quad n=\frac{10}{4}=\frac{5}{2} \quad \text{or} \quad n=\frac{-8}{4}=-2. \] Therefore, the numerator factors (with a suitable constant) as \[ 2n^2 - n -10 = 2\Bigl(n-\frac{5}{2}\Bigr)(n+2). \] (One may check by expanding that this factorization is correct.) 3. **Cancel the common factor (with a restriction).** For \( n\neq -2 \) the factor \( (n+2) \) may be cancelled. (Indeed, if \( n=-2 \) then the denominator is zero.) Thus, for \( n\neq -2 \) we have \[ \frac{2n^{2}-n-10}{2n^{2}+3n-2} = \frac{2\Bigl(n-\frac{5}{2}\Bigr)(n+2)}{(2n-1)(n+2)} = \frac{2\Bigl(n-\frac{5}{2}\Bigr)}{2n-1}. \] 4. **Rewrite in a simpler form.** Write \[ n-\frac{5}{2}=\frac{2n-5}{2}. \] Then \[ \frac{2\Bigl(n-\frac{5}{2}\Bigr)}{2n-1}=\frac{2\cdot\frac{2n-5}{2}}{2n-1}=\frac{2n-5}{2n-1}. \] Hence, for \( n\in\mathbb{Z}\setminus\{-2\} \) \[ \frac{2n^{2}-n-10}{2n^{2}+3n-2}=\frac{2n-5}{2n-1}. \] 5. **A convenient re‐parameterization.** Notice that we can write \[ \frac{2n-5}{2n-1} = \frac{(2n-1)-4}{2n-1}= 1-\frac{4}{2n-1}. \] Since when \( n \) runs over all integers (except \( n=-2 \) to avoid \(2n-1=-5\) which came from the cancellation) the quantity \( k=2n-1 \) runs over all odd integers except \( k=-5 \). Thus, one acceptable answer is: \[ \boxed{A_3=\left\{1-\frac{4}{k}: k\in\{\, \text{all odd integers}\,\}\setminus\{-5\}\right\}.} \] --- **(b) Description of** \[ A_{4}=\{x\in\mathbb{Q}: x\in(0,6]\}. \] This set is simply the collection of all rational numbers that lie in the interval \( (0,6] \). It is a dense subset (with respect to the usual topology on \(\mathbb{R}\)) of the interval \( (0,6] \); that is, between any two real numbers in \( (0,6] \) there is a rational number. In boxed form, one may write: \[ \boxed{A_{4}=\{x\in\mathbb{Q}: 0