(a) To find the probability distribution table for X, we first list all possible outcomes of the sums of the numbers from the two spinners:

• Four-sided spinner outcomes: 1, 2, 2, 3
• Three-sided spinner outcomes: -2, -1, 1

Possible sums (X):

• 1 + (-2) = -1
• 1 + (-1) = 0
• 1 + 1 = 2
• 2 + (-2) = 0
• 2 + (-1) = 1
• 2 + 1 = 3
• 3 + (-2) = 1
• 3 + (-1) = 2
• 3 + 1 = 4

Count the occurrences of each sum and divide by the total number of outcomes (12) to get the probabilities:

• \(P(X = -1) = \frac{1}{12}\)
• \(P(X = 0) = \frac{3}{12}\)
• \(P(X = 1) = \frac{3}{12}\)
• \(P(X = 2) = \frac{2}{12}\)
• \(P(X = 3) = \frac{2}{12}\)
• \(P(X = 4) = \frac{1}{12}\)

(b) To find \(\text{Var}(X)\), we first calculate \(\text{E}(X)\):

\(\text{E}(X) = \frac{-1 + 0 + 3 + 4 + 6 + 4}{12} = \frac{16}{12} = \frac{4}{3}\)

Then, calculate \(\text{E}(X^2)\):

\(\text{E}(X^2) = \frac{1 + 0 + 3 + 8 + 18 + 16}{12} = \frac{46}{12}\)

Finally, \(\text{Var}(X) = \text{E}(X^2) - (\text{E}(X))^2\):

\(\text{Var}(X) = \frac{46}{12} - \left(\frac{4}{3}\right)^2 = \frac{37}{18} \approx 2.06\)