a) \( (2 p-3 q)(p-q)(p+q)\left(4 p^{2}+6 p q+9 q^{2}\right) \)

To expand and simplify the expression: \[ (2p - 3q)(p - q)(p + q)\left(4p^{2} + 6pq + 9q^{2}\right) \] Let's follow these steps: ### Step 1: Simplify \((p - q)(p + q)\) \[ (p - q)(p + q) = p^2 - q^2 \] ### Step 2: Multiply by \((2p - 3q)\) Now the expression becomes: \[ (2p - 3q)(p^2 - q^2)\left(4p^{2} + 6pq + 9q^{2}\right) \] ### Step 3: Recognize Patterns for Further Simplification Notice that \(4p^2 + 6pq + 9q^2\) is closely related to the expansion of \((2p + 3q)^2\): \[ (2p + 3q)^2 = 4p^2 + 12pq + 9q^2 \] However, this doesn't directly factor into our expression. Instead, observe that the entire expression can be further factored by recognizing it as part of a difference of cubes: \[ 8p^3 - 27q^3 = (2p)^3 - (3q)^3 = (2p - 3q)(4p^2 + 6pq + 9q^2) \] So the original expression can be rewritten as: \[ (p - q)(8p^3 - 27q^3) \] ### Step 4: Expand the Expression Fully To present the expression in its expanded form: \[ (p - q)(8p^3 - 27q^3) = 8p^4 - 8p^3q - 27pq^3 + 27q^4 \] ### Final Answer \[ 8p^{4} - 8p^{3}q - 27p q^{3} + 27q^{4} \]