a) \( A_{1}=\left\{x \in \mathbb{R}: 2 x^{2}+5 x-4<-1\right\} \) b) \( A_{2}=\{n \in \mathbb{N}: 3+2|4 n-3|>13\} \)

**a) Solve \(2x^2+5x-4<-1\):** 1. Rewrite the inequality: \[ 2x^2+5x-4 < -1 \] Add \(1\) to both sides: \[ 2x^2+5x-3 < 0 \] 2. Find the roots of the quadratic equation: \[ 2x^2+5x-3=0 \] Compute the discriminant: \[ \Delta = 5^2 - 4(2)(-3) = 25 + 24 = 49 \] The roots are: \[ x = \frac{-5 \pm \sqrt{49}}{2 \cdot 2} = \frac{-5 \pm 7}{4} \] This gives: \[ x_1 = \frac{-5+7}{4} = \frac{2}{4} = \frac{1}{2}, \quad x_2 = \frac{-5-7}{4} = \frac{-12}{4} = -3 \] 3. Determine the interval where the quadratic is negative. Since the coefficient of \(x^2\) (which is \(2\)) is positive, the quadratic is negative between its roots: \[ -3 < x < \frac{1}{2} \] Thus, \[ A_{1} = \{x \in \mathbb{R} : -3 < x < \frac{1}{2}\}. \] --- **b) Solve \(3+2|4n-3|>13\) for \(n\in\mathbb{N}\):** 1. Start with the inequality: \[ 3+2|4n-3| > 13 \] Subtract \(3\) from both sides: \[ 2|4n-3| > 10 \] Divide both sides by \(2\): \[ |4n-3| > 5 \] 2. Solve the absolute value inequality which splits into two cases: **Case 1:** \[ 4n-3 > 5 \quad \Rightarrow \quad 4n > 8 \quad \Rightarrow \quad n > 2 \] **Case 2:** \[ 4n-3 < -5 \quad \Rightarrow \quad 4n < -2 \quad \Rightarrow \quad n < -\frac{1}{2} \] Since \(n\) is a natural number, the second case gives no solutions. 3. From Case 1, the natural numbers satisfying \(n > 2\) are: \[ n = 3, 4, 5, \dots \] Thus, \[ A_{2} = \{n \in \mathbb{N} : n \geq 3\}. \]