\( \left. \begin{array} { l l } { 6.16 } & { ( x - 3 ) ( x + 3 ) + ( 2 x - 7 ) ( x - 1 ) - ( x - 5 ) ( x + 2 ) } \\ { 6.17 } & { - 3 a b ( 2 a - b ) ( 2 a + b ) } \\ { 6.18 } & { ( 2 x - 3 ) ( 4 x ^ { 2 } + 6 x + 9 ) } \end{array} \right. \)

Below is the step‐by‐step solution for each item. --- ### Problem 6.16 Simplify \[ ( x - 3 ) ( x + 3 ) + ( 2x - 7 ) ( x - 1 ) - ( x - 5 ) ( x + 2 ). \] 1. **Expand the products:** - For \((x-3)(x+3)\): \[ (x-3)(x+3) = x^2 - 9. \] - For \((2x-7)(x-1)\): \[ (2x-7)(x-1) = 2x \cdot x - 2x \cdot 1 - 7 \cdot x + 7 = 2x^2 - 9x + 7. \] - For \((x-5)(x+2)\): \[ (x-5)(x+2) = x^2 + 2x - 5x - 10 = x^2 - 3x - 10. \] 2. **Combine the expanded expressions:** Substitute into the original expression: \[ \begin{aligned} &\quad (x^2 - 9) + (2x^2 - 9x + 7) - (x^2 - 3x - 10) \\ &= x^2 - 9 + 2x^2 - 9x + 7 - x^2 + 3x + 10. \end{aligned} \] 3. **Combine like terms:** - \(x^2\) terms: \[ x^2 + 2x^2 - x^2 = 2x^2. \] - \(x\) terms: \[ -9x + 3x = -6x. \] - Constant terms: \[ -9 + 7 + 10 = 8. \] So, the expression simplifies to: \[ 2x^2 - 6x + 8. \] 4. **Optional factorization:** We can factor out a common factor \(2\): \[ 2(x^2 - 3x + 4). \] Hence, the simplified form is: \[ \boxed{2x^2 - 6x + 8 \quad \text{or} \quad 2(x^2-3x+4)}. \] --- ### Problem 6.17 Simplify \[ -3ab(2a-b)(2a+b). \] 1. **Recognize the difference of squares:** The product \((2a-b)(2a+b)\) is of the form \((A-B)(A+B)=A^2-B^2\) with \(A=2a\) and \(B=b\). Thus, \[ (2a-b)(2a+b)= (2a)^2 - b^2 = 4a^2 - b^2. \] 2. **Combine with the remaining factor:** Multiply by \(-3ab\): \[ -3ab(4a^2 - b^2). \] The final simplified form is: \[ \boxed{-3ab(4a^2 - b^2)}. \] --- ### Problem 6.18 Simplify \[ (2x-3)(4x^2+6x+9). \] 1. **Multiply the factors:** Distribute \((2x-3)\) over the trinomial: \[ \begin{aligned} (2x-3)(4x^2+6x+9) &= 2x(4x^2+6x+9) - 3(4x^2+6x+9) \\ &= 8x^3 + 12x^2 + 18x - 12x^2 - 18x - 27. \end{aligned} \] 2. **Combine like terms:** Notice that the \(x^2\) and \(x\) terms cancel: \[ 8x^3 - 27. \] 3. **Recognize the difference of cubes:** The result can be seen as a difference of cubes: \[ 8x^3 - 27 = (2x)^3 - 3^3, \] which factors back into: \[ (2x-3)(4x^2+6x+9), \] confirming our work. Thus, the simplified form is: \[ \boxed{8x^3 - 27}. \] --- These are the step-by-step solutions for all three problems.