Iatch each expression on the left with its product on the right. \( \begin{array}{ll}(3 d+1)\left(d^{2}-2\right) & 3 d^{3}+6 d^{2}-d- \\ (d-1)\left(3 d^{2}+2 d+2\right) & 3 d^{3}+d^{2}-6 d- \\ \left(3 d^{2}-1\right)(d+2) & 3 d^{3}-d^{2}-2\end{array} \)

To match each expression on the left with its product on the right, we will expand each expression and then compare the results with the products given. 1. **Expand \( (3d + 1)(d^2 - 2) \)**: \[ (3d + 1)(d^2 - 2) = 3d \cdot d^2 + 3d \cdot (-2) + 1 \cdot d^2 + 1 \cdot (-2) \] \[ = 3d^3 - 6d + d^2 - 2 \] Rearranging gives: \[ = 3d^3 + d^2 - 6d - 2 \] 2. **Expand \( (d - 1)(3d^2 + 2d + 2) \)**: \[ (d - 1)(3d^2 + 2d + 2) = d \cdot 3d^2 + d \cdot 2d + d \cdot 2 - 1 \cdot 3d^2 - 1 \cdot 2d - 1 \cdot 2 \] \[ = 3d^3 + 2d^2 + 2d - 3d^2 - 2d - 2 \] Combining like terms gives: \[ = 3d^3 - d^2 - 2 \] 3. **Expand \( (3d^2 - 1)(d + 2) \)**: \[ (3d^2 - 1)(d + 2) = 3d^2 \cdot d + 3d^2 \cdot 2 - 1 \cdot d - 1 \cdot 2 \] \[ = 3d^3 + 6d^2 - d - 2 \] Now we can match the expanded products with the given products: - \( (3d + 1)(d^2 - 2) \) gives \( 3d^3 + d^2 - 6d - 2 \) - \( (d - 1)(3d^2 + 2d + 2) \) gives \( 3d^3 - d^2 - 2 \) - \( (3d^2 - 1)(d + 2) \) gives \( 3d^3 + 6d^2 - d - 2 \) Now we can match them: - \( (3d + 1)(d^2 - 2) \) matches with \( 3d^3 + d^2 - 6d - 2 \) - \( (d - 1)(3d^2 + 2d + 2) \) matches with \( 3d^3 - d^2 - 2 \) - \( (3d^2 - 1)(d + 2) \) matches with \( 3d^3 + 6d^2 - d - 2 \) Thus, the final matches are: \[ \begin{array}{ll} (3d + 1)(d^2 - 2) & 3d^3 + d^2 - 6d - 2 \\ (d - 1)(3d^2 + 2d + 2) & 3d^3 - d^2 - 2 \\ (3d^2 - 1)(d + 2) & 3d^3 + 6d^2 - d - 2 \\ \end{array} \]