\( ( - 7 q ^ { 6 } - q ^ { 4 } ) ( - 2 q ^ { 2 } - q ^ { 5 } ) \)

To multiply the two polynomials, use the distributive property (also known as the FOIL method for binomials): First, distribute \( -7q^6 \) to both terms in the second polynomial: \[ -7q^6 \cdot (-2q^2) = 14q^8 \] \[ -7q^6 \cdot (-q^5) = 7q^{11} \] Next, distribute \( -q^4 \) to both terms in the second polynomial: \[ -q^4 \cdot (-2q^2) = 2q^6 \] \[ -q^4 \cdot (-q^5) = q^9 \] Now, combine all the resulting terms: \[ 7q^{11} + 14q^8 + q^9 + 2q^6 \] Rearranging the terms in descending order gives: \[ 7q^{11} + 14q^8 + q^9 + 2q^6 \] Combining like terms results in: \[ 7q^{11} + q^9 + 14q^8 + 2q^6 \] Thus, the final product is: \[ 7q^{11} + q^9 + 14q^8 + 2q^6 \]