1. The problem asks to determine the truth value of several algebraic and geometric statements. 2. Statement 1: For any integers $a$, $b$, and $c$, $a(b + c) = ab + ac$. This is the distributive property of multiplication over addition, which is true. 3. Statement 2: An algebraic expression $2x^2 + 2xy$ is a trinomial. A trinomial has three terms; here there are only two terms, so this is false. 4. Statement 3: $3a^2b$ and $3ab^2$ are like terms. Like terms have the same variables raised to the same powers. Here, powers differ, so false. 5. Statement 4: If $a = b$, then $a - b = 0$. Substituting $a$ for $b$ gives $a - a = 0$, true. 6. Statement 5: The equation of the line $x + y = 0$ is a vertical line. Rearranged: $y = -x$, which is not vertical, so false. 7. Statement 6: $4$ is the slope of the line $y - 4x = 0$. Rearranged: $y = 4x$, slope is $4$, true. 8. Statement 7: The numerical coefficient of $3xy^2$ is $2$. The coefficient is $3$, so false. 9. Statement 8: If $c = d$, then $a + c = d + a$. By commutative property, true. Final answers: 1. True 2. False 3. False 4. True 5. False 6. True 7. False 8. True