1. 5(x+7)+(x-6)(x+6) 2. (3a-2)(3a+2)-(1+a)(a-1) 3. (x+3)2 - (x -4)2 4. (2y + 3) (2y - 3) - 3y2 5. (a - 3) (3 + a) + (a - 3)2 6. (x + 3) (x - 3) - 82 + 2x 7. (a - 3)2 - а(а - 3)…

Let’s solve each of the expressions step by step:

1. \(5(x+7)+(x-6)(x+6)\)

   • First, simplify the expression: \[ 5(x + 7) = 5x + 35 \] \[ (x - 6)(x + 6) = x^2 - 36 \quad \text{(Difference of squares)} \]
   • Combine everything: \[ 5x + 35 + x^2 - 36 = x^2 + 5x - 1 \]
   • Final answer: \(x^2 + 5x - 1\)

2. \((3a-2)(3a+2)-(1+a)(a-1)\)

   • Use the difference of squares: \[ (3a-2)(3a+2) = 9a^2 - 4 \] \[ (1+a)(a-1) = a^2 - 1 \]
   • Combine: \[ 9a^2 - 4 - (a^2 - 1) = 9a^2 - 4 - a^2 + 1 = 8a^2 - 3 \]
   • Final answer: \(8a^2 - 3\)

3. \((x+3)^2 - (x - 4)^2\)

   • Use the difference of squares: \[ (x+3 - (x-4))(x+3 + (x-4)) = (7)(2x - 1) \]
   • Final answer: \(14x - 7\)

4. \((2y + 3)(2y - 3) - 3y^2\)

   • Use the difference of squares: \[ (2y + 3)(2y - 3) = 4y^2 - 9 \]
   • Combine: \[ 4y^2 - 9 - 3y^2 = y^2 - 9 \]
   • Final answer: \(y^2 - 9\)

5. \((a - 3)(3 + a) + (a - 3)^2\)

   • Expand the first part: \[ (a - 3)(3 + a) = a^2 + 3a - 3a - 9 = a^2 - 9 \]
   • Expand the second part: \[ (a - 3)^2 = a^2 - 6a + 9 \]
   • Combine: \[ (a^2 - 9) + (a^2 - 6a + 9) = 2a^2 - 6a \]
   • Final answer: \(2a^2 - 6a\)

6. \((x + 3)(x - 3) - 82 + 2x\)

   • Use the difference of squares: \[ (x + 3)(x - 3) = x^2 - 9 \]
   • Combine: \[ x^2 - 9 - 82 + 2x = x^2 + 2x - 91 \]
   • Final answer: \(x^2 + 2x - 91\)

7. \((a - 3)^2 - a(a - 3)\) when \(a = 1.7\)

   • First, calculate \((a - 3)^2\): \[ (1.7 - 3)^2 = (-1.3)^2 = 1.69 \]
   • Then calculate \(a(a - 3)\): \[ 1.7(1.7 - 3) = 1.7(-1.3) = -2.21 \]
   • Finally combine: \[ 1.69 - (-2.21) = 1.69 + 2.21 = 3.90 \]
   • Final answer: \(3.90\)

8. \(2(3x + 1)^2\) when \(x = -3\)

   • First, calculate \(3x + 1\): \[ 3(-3) + 1 = -9 + 1 = -8 \]
   • Now calculate \((3x + 1)^2\): \[ (-8)^2 = 64 \]
   • Finally multiply: \[ 2 \times 64 = 128 \]
   • Final answer: \(128\)

You can use these solutions to study or review the material for your exams! If you have any more questions or need further assistance, feel free to ask!