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Question 1: Simplify and identify the equivalent expression.

Expression: $-3a(a + b - 5) + 4(-2a + 2b) + b(a + 3b - 7)$

Step-by-step simplification:

1. Expand each term:

   • $-3a(a + b - 5) = -3a^2 - 3ab + 15a$
   • $4(-2a + 2b) = -8a + 8b$
   • $b(a + 3b - 7) = ab + 3b^2 - 7b$

2. Combine all terms: $-3a^2 - 3ab + 15a - 8a + 8b + ab + 3b^2 - 7b$

3. Group like terms:

   • For $a^2$: $-3a^2$
   • For $ab$: $-3ab + ab = -2ab$
   • For $a$: $15a - 8a = 7a$
   • For $b^2$: $3b^2$
   • For $b$: $8b - 7b = b$

Final simplified expression: $-3a^2 + 3b^2 - 2ab + 7a + b$

Answer: $D$ $-3a^2 + 3b^2 - 2ab + 7a + b$.

Question 2: Multiply the polynomials $(x + 3)(2x - 4)$.

Step-by-step multiplication:

1. Use the distributive property: $(x + 3)(2x - 4) = x(2x - 4) + 3(2x - 4)$

2. Expand each term:

   • $x(2x - 4) = 2x^2 - 4x$
   • $3(2x - 4) = 6x - 12$

3. Combine like terms: $2x^2 - 4x + 6x - 12 = 2x^2 + 2x - 12$

Answer: $a = 2, \, b = 2, \, c = -12$

Question 3: Simplify $3 + 2(x + 4)(x - 4)$.

Step-by-step simplification:

1. Recognize the difference of squares: $(x + 4)(x - 4) = x^2 - 16$

2. Substitute into the expression: $3 + 2(x^2 - 16)$

3. Distribute the $2$: $3 + 2x^2 - 32$

4. Combine constants: $2x^2 - 29$

Answer: $B$ $2x^2 - 29$.

Let me know if you'd like detailed explanations for any step or have further questions! Here are five questions to expand your understanding:

1. How can the distributive property simplify polynomial multiplication?
2. What are the steps to identify like terms when combining expressions?
3. How is the difference of squares formula applied in simplifying polynomials?
4. How does grouping terms help in simplifying large algebraic expressions?
5. Can you identify and explain when to use factoring in similar problems?

Tip: Always double-check signs when expanding or combining terms to avoid errors!