For x -2, which expression is equivalent to x^2 + 4x - 1x + 2? | SAT Equivalent expressions Question, Answer Explained

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Math · Advanced MathMedium

For $x \ne -2$ , which expression is equivalent to $\frac{x^2 + 4x - 1}{x + 2}$ ?

$x + 2 - \frac{5}{x + 2}$

$x + 2 - \frac{1}{x + 2}$

$x + 2 + \frac{3}{x + 2}$

$x + 4 - \frac{1}{x + 2}$

Show answer and explanation

Answer: A

Since $(x + 2)^2 = x^2 + 4x + 4$ , the numerator is $x^2 + 4x - 1 = (x + 2)^2 - 5$ . Dividing each part by $x + 2$ gives $\frac{(x + 2)^2}{x + 2} - \frac{5}{x + 2} = x + 2 - \frac{5}{x + 2}$ .

Choice B keeps the $-1$ from the numerator as the remainder, but $(x + 2)^2$ already uses $+4$ , so the remainder is $-1 - 4 = -5$ .

Why the other choices are wrong

B. This uses -1 as the remainder. Since (x + 2)² = x² + 4x + 4, the remainder is -1 - 4 = -5.
C. This adds 4 to -1 instead of subtracting 4. The remainder is -1 - 4 = -5.
D. This divides x² + 4x by x instead of by x + 2, then puts the -1 over x + 2.

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