Step-by-step Solution:

Let the first term of the arithmetic progression be \( a \) and the given common difference be \( d = \frac{3}{2} \). The sum of the first 11 terms is given to be 88: \[ S_{11} = \frac{11}{2} (2a + 10d) = 88 \] \[ 2a + 10\left(\frac{3}{2}\right) = 16 \] \[ 2a + 15 = 16 \implies 2a = 1 \implies a = \frac{1}{2} \] The roots of the quadratic equation correspond to the 10th and 11th terms: \[ T_{10} = a + 9d = \frac{1}{2} + 9\left(\frac{3}{2}\right) = \frac{28}{2} = 14 \] \[ T_{11} = a + 10d = \frac{1}{2} + 10\left(\frac{3}{2}\right) = \frac{31}{2} = 15.5 \] For the quadratic equation \( 3x^2 - px + q = 0 \), the sum and product of the roots correlate to its coefficients: \[ \text{Sum of roots} = \frac{p}{3} = 14 + 15.5 = 29.5 \] \[ p = 3 \times 29.5 = 88.5 = \frac{177}{2} \] \[ \text{Product of roots} = \frac{q}{3} = 14 \times \frac{31}{2} = 217 \] \[ q = 3 \times 217 = 651 \] We need to compute the specific value of \( q - 2p \): \[ q - 2p = 651 - 2\left(\frac{177}{2}\right) = 651 - 177 = 474 \]