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The image contains various logic and polynomial problems. Here’s a breakdown of the questions:

### 2. Logic Propositions
If \( p \) and \( r \) are true, and \( q \) is false, determine the truth values of the following:
- a) \( [(p \land \neg q) \lor \neg r] \Rightarrow q \)
- b) \( [(\neg r \lor q) \land (r \lor \neg p)] \Leftrightarrow \neg \neg r \)
- c) \( [[\neg p \Rightarrow q] \Leftrightarrow \neg r] \lor [\neg q \Rightarrow r] \)

### 3. Logic Proposition Truth Values
Given that the proposition \( [(p \lor q) \land p] \Rightarrow [(r \lor q) \Leftrightarrow p] \) is false, determine the truth values of the propositions:
- a) \( p, q, r \)
- b) \( [(p \land \neg q) \Rightarrow (r \lor p)] \Leftrightarrow [\neg q \land (r \lor p)] \)

### 4. Polynomial Problems
- a) Perform the addition, subtraction, and multiplication of the polynomials \( P(x) = 5x^3 - x + 7x^3 - x^2 + 8x - 2 \) and \( Q(x) = 12 + x^2 + 7x - x^4 - 8 - 3x^2 \).
- b) Expand and simplify \( (3x + 2)^2 \).

Would you like me to work through the solutions for these problems step by step, starting with a specific one?

Logic Propositions and Polynomial Operations: If p and r are true, and q is false, determine the truth values of various propositions. Additionally, solve the polynomial operations for P(x) = 5x^3 - x + 7x^3 - x^2 + 8x - 2 and Q(x) = 12 + x^2 + 7x - x^4 - 8 - 3x^2.

Solve logic proposition truth values for p, q, and r using Boolean algebra, and perform polynomial operations for P(x) and Q(x). Includes step-by-step solutions for truth tables and polynomial arithmetic.

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