Question

which graph represents the solution set for the quadratic inequality \\(x^2 + 2x + 1 > 0\\)?

Explanation:

Factor the quadratic expression

$$ x^2 + 2x + 1 = (x + 1)^2 $$

Solve the inequality

$$ (x + 1)^2 > 0 $$

$$ x + 1 eq 0 \implies x eq -1 $$

Identify the corresponding graph

The solution set is all real numbers except \(x = -1\), which is represented on a number line by shading everywhere except for an open circle at \(-1\). This matches the third graph.

Answer:

• (A) Shaded region to the right of \(-1\) with a solid point at \(-1\)
• (B) Shaded region to the left of \(-1\) with a solid point at \(-1\)
• (C) Shaded region in both directions with an open circle at \(-1\) (Correct answer)
• (D) Shaded region to the right of \(1\) with an open circle at \(1\)