Distribute the negative sign to both terms in the parentheses: 8 - (2x - 3) = 8 - 2x + 3. Combine the constant terms: 8 + 3 = 11, giving 8 - 2x + 3 = 11 - 2x. The simplified form is 11 - 2x, which equals -2x + 11.

The correct answer is B (5x − 12). Distribute the 3 across the parentheses: 3(x − 4) = 3x − 12. Then combine like terms with 2x: 3x − 12 + 2x = 5x − 12. A (5x − 4) results from distributing 3 to x but not to −4, keeping −4 instead of computing 3 × (−4) = −12. C (6x − 12) comes from incorrectly treating the 2x as adding to the coefficient 3 rather than to 3x. D (x − 12) results from subtracting 2x rather than adding it. Distribution errors are extremely common — always multiply the outside factor by every term inside the parentheses.

The correct answer is C (12x⁴y⁶). Multiply the coefficients: 3 × 4 = 12. Apply the product rule for exponents (add exponents of like bases): x³ × x¹ = x^(3+1) = x⁴; y² × y⁴ = y^(2+4) = y⁶. Result: 12x⁴y⁶. A (7x³y⁶) adds coefficients (3+4=7) instead of multiplying and doesn't add the x exponent. B (12x³y⁸) correctly multiplies coefficients but treats x³ as fixed and incorrectly adds the y exponents (possibly counting y⁴ twice). D (7x⁴y⁶) adds coefficients but correctly adds the exponents. Key rule: multiply coefficients, add exponents of like bases.

This is a difference of squares pattern: (a + b)(a - b) = a² - b². Apply this with a = x and b = 2: (x + 2)(x - 2) = x² - 2² = x² - 4. The middle terms cancel when expanding: x² - 2x + 2x - 4 = x² - 4. Choice B incorrectly shows +4 instead of -4.

To simplify this expression, we need to first simplify inside the parentheses, then apply the distributive property. Inside the parentheses: 2 + 5 = 7. The expression becomes 3x(7) = 21x. We can also think of this as distributing: 3x(2 + 5) = 3x(2) + 3x(5) = 6x + 15x = 21x. Choice A incorrectly leaves the expression in distributed form without combining like terms.

Apply the distributive property with -3 multiplied by each term inside the parentheses. -3 × 2x = -6x and -3 × (-5) = +15 (negative times negative equals positive). Therefore, -3(2x - 5) = -6x + 15. Choice A incorrectly shows -15 instead of +15.

To simplify this expression, we need to apply the distributive property. First, distribute the -3 to both terms in the parentheses: -3(2x - 4) = -6x + 12. The expression becomes 7x - 6x + 12. Combining like terms: (7x - 6x) + 12 = x + 12. Choice C incorrectly adds the coefficients of x terms instead of subtracting.

First distribute x through the parentheses: x(x - 7) = x² - 7x. Then add 3x to get: x² - 7x + 3x = x² - 4x. Combine like terms: -7x + 3x = -4x, while the x² term remains unchanged. Choice B incorrectly shows -10x instead of -4x.

Multiply the coefficients and add the exponents when multiplying powers with the same base. 2x² × 3x = (2 × 3)(x² × x) = 6x³. The exponents add: x² × x¹ = x²⁺¹ = x³. Choice A incorrectly multiplies the exponents instead of adding them.

To factor this quadratic expression, find two numbers that multiply to -5 (the constant term) and add to -4 (the coefficient of x). The numbers -5 and 1 satisfy both conditions: (-5) × 1 = -5 and (-5) + 1 = -4. Therefore, x² - 4x - 5 = (x - 5)(x + 1). You can verify by expanding: (x - 5)(x + 1) = x² + x - 5x - 5 = x² - 4x - 5.