Algebraic expressions

Correct answer: B. 21 - 7x

Explanation: The governing concept is distribution of a product and correct removal of an outer negative sign. First evaluate the inner product: 3(2 − x) = 6 − 3x. The expression inside the square brackets then becomes 4x − (6 − 3x) = 4x − 6 + 3x = 7x − 6. Substitute this into the original expression: 15 − (7x − 6). The minus sign applies to both terms in the bracket, giving 15 − 7x + 6 = 21 − 7x. Therefore option B is correct. Option A loses the coefficient 7 produced by combining 4x and 3x. Options C and D result from mishandling the outer subtraction or changing only one sign instead of both. The signs and like terms must be handled together.

Correct answer: D. (x − y)² + 4xy

Explanation: The governing concept is translating a compound verbal statement into algebra while respecting grouping. “The difference of x and y” is x − y. Because the statement asks for “the square of the difference,” the entire difference must be enclosed in parentheses before squaring: (x − y)². Adding 4xy to this quantity gives (x − y)² + 4xy, so option D is correct. Option A uses x² − y², which is the difference of two squares and is not the square of a difference. Option B squares only y and therefore does not represent the stated phrase. Option C subtracts the square from 4xy, reversing the requested addition and order. Parentheses are essential here.

Correct answer: A. \(x^2+x+1\)

Explanation: Factor the numerator: \(x^2-1=(x-1)(x+1)\). Since \(x\neq1\), \(\frac{x^2-1}{x-1}=x+1\). Hence the given expression becomes \(x^2+(x+1)=x^2+x+1\), so option A is correct. Option B incorrectly has a constant term of \(-1\); the simplified expression has \(+1\). Exam tip: Factor the numerator fully before cancelling a common factor, and retain the condition that the denominator must not be zero.