A function \( \mathrm{g}(\mathrm{x})=\mathrm{x}+2 \) and \( \mathrm{h}(\mathrm{x})=x^{2} \). Find the value of x for which \( \mathrm{hg}(\mathrm{x})=3 x^{2}+x-1 \) Select one: a. \( \quad x=\frac{-5}{2}, 1 \) b. \( \quad x=\frac{5}{2}, 1 \) c. \( \quad x=\frac{-5}{2},-1 \) d. \( \quad x=\frac{5}{2},-1 \)

We are given the functions \[ \mathrm{g}(\mathrm{x}) = \mathrm{x} + 2 \quad \text{and} \quad \mathrm{h}(\mathrm{x}) = \mathrm{x}^2. \] The composition is \[ \mathrm{hg}(\mathrm{x}) = \mathrm{h}(\mathrm{g}(\mathrm{x})) = \big(\mathrm{g}(\mathrm{x})\big)^2 = (\mathrm{x} + 2)^2. \] We need to find \( x \) such that \[ (\mathrm{x} + 2)^2 = 3\mathrm{x}^2 + \mathrm{x} - 1. \] **Step 1. Expand the left-hand side:** \[ (\mathrm{x} + 2)^2 = \mathrm{x}^2 + 4\mathrm{x} + 4. \] **Step 2. Set up the equation:** \[ \mathrm{x}^2 + 4\mathrm{x} + 4 = 3\mathrm{x}^2 + \mathrm{x} - 1. \] **Step 3. Rearrange the equation to form a quadratic equation:** Subtract \(\mathrm{x}^2 + 4\mathrm{x} + 4\) from both sides: \[ 0 = 3\mathrm{x}^2 + \mathrm{x} - 1 - (\mathrm{x}^2 + 4\mathrm{x} + 4). \] Simplify: \[ 0 = (3\mathrm{x}^2 - \mathrm{x}^2) + (\mathrm{x} - 4\mathrm{x}) + (-1 - 4), \] \[ 0 = 2\mathrm{x}^2 - 3\mathrm{x} - 5. \] **Step 4. Solve the quadratic equation \( 2\mathrm{x}^2 - 3\mathrm{x} - 5 = 0 \):** The quadratic formula is given by \[ \mathrm{x} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \] where \( a = 2 \), \( b = -3 \), and \( c = -5 \). First, calculate the discriminant: \[ \Delta = b^2 - 4ac = (-3)^2 - 4(2)(-5) = 9 + 40 = 49. \] Now, compute \( x \): \[ \mathrm{x} = \frac{-(-3) \pm \sqrt{49}}{2 \cdot 2} = \frac{3 \pm 7}{4}. \] This gives two solutions: \[ \mathrm{x} = \frac{3 + 7}{4} = \frac{10}{4} = \frac{5}{2}, \] \[ \mathrm{x} = \frac{3 - 7}{4} = \frac{-4}{4} = -1. \] **Step 5. Identify the correct answer:** The two solutions are \( \mathrm{x} = \frac{5}{2} \) and \( \mathrm{x} = -1 \). Thus, the correct option is: \[ \text{d. } \quad x = \frac{5}{2}, -1. \]