C. \( \frac{1}{2}\left(12 x^{2}+2\right)-\left(10 x^{2}-3 x-1\right) \)

To simplify the expression \( \frac{1}{2}\left(12 x^{2}+2\right)-\left(10 x^{2}-3 x-1\right) \), start by distributing the \( \frac{1}{2} \): \[ \frac{1}{2} \cdot 12x^2 + \frac{1}{2} \cdot 2 = 6x^2 + 1. \] Now, rewrite the full expression: \[ 6x^2 + 1 - (10x^2 - 3x - 1). \] Distributing the negative sign gives us: \[ 6x^2 + 1 - 10x^2 + 3x + 1. \] Now, combine like terms: - For \( x^2 \): \( 6x^2 - 10x^2 = -4x^2 \), - For \( x \): \( 3x \), - For constants: \( 1 + 1 = 2 \). Putting it all together, the simplified expression is: \[ -4x^2 + 3x + 2. \] So the final answer is: \[ -4x^2 + 3x + 2. \]