\[ 12 \text { manmen } a(x-2)(x+4) \] In the quadratic equation abovc, \( a \) is a nonzero constant. The graph of the equation in the \( x y \)-plane is a parabola with vertex \( (c, d) \). Which of the following is equal to a ? A) \( -9 a \) B) \( -8 a \) C) \( -5 a \) D) \( -2 a \)

To find the value of \( a \) in the quadratic expression \( 12 \text{ manmen } a(x-2)(x+4) \), we need to understand the standard form of a quadratic equation and how to extract the coefficient of \( x^2 \). First, let's rewrite the equation in standard form. The expression \( (x-2)(x+4) \) can be expanded: \[ (x-2)(x+4) = x^2 + 4x - 2x - 8 = x^2 + 2x - 8. \] So, the quadratic becomes: \[ 12 \text{ manmen } a \cdot (x^2 + 2x - 8) = 12a x^2 + 24a x - 96a. \] In this case, the coefficient of \( x^2 \) is \( 12a \). The question asks for a value of \( a \) that satisfies one of the options. Rearranging might provide insight into what they represent: - A) \( -9a \) - B) \( -8a \) - C) \( -5a \) - D) \( -2a \) Given the structure of the problem, it seems likely that each option expresses a relationship involving \( a \). ### Finding \( a \): To determine which option equals \( a \), we can solve for each: 1. **A:** If \( 12a = -9a \), then \( 12a + 9a = 0 \) → \( 21a = 0 \) → no viable solution with \( a \neq 0\). 2. **B:** If \( 12a = -8a \), then \( 12a + 8a = 0 \) → \( 20a = 0 \) → no viable solution with \( a \neq 0\). 3. **C:** If \( 12a = -5a \), then \( 12a + 5a = 0 \) → \( 17a = 0 \) → no viable solution with \( a \neq 0\). 4. **D:** If \( 12a = -2a \), then \( 12a + 2a = 0 \) → \( 14a = 0 \) → no viable solution with \( a \neq 0\). In this case, it appears all options do not directly provide a conflicting representation with non-zero \( a \). However, in most contexts, one would conclude after evaluating all expressions that \( a \) should be positive and potentially \( -2 \) is the solution that keeps its viability in check against typical formats of quadratic equations; ultimately it simplifies to: Final decision: Always verify with more context or further numerical examples if actual problems yield consistent viability checks. So the correct \( a \) in most evaluative context of these type expressions ends up leaning towards **D)** simplest yield compliance that \( a = -2 \).