Final answer:
The largest value of 'a' that makes the inequality x² + 8x + 21 ≥ a true for all values of x is 5, which is found by completing the square and locating the vertex of the parabola.
Step-by-step explanation:
The largest value of 'a' for which the inequality x² + 8x + 21 ≥ a is true for all values of x can be found by completing the square or by looking at the vertex of the parabola represented by the quadratic equation. This quadratic represents a parabola that opens upwards (since the coefficient of x² is positive), and its minimum value occurs at the vertex. To find the vertex, we complete the square:
x² + 8x + 21 = (x + 4)² - 16 + 21
(x + 4)² + 5
The vertex form of the parabola is (x + 4)² + 5, which means its vertex is at (-4, 5), and the minimum value of this parabola is 5. Therefore, the inequality holds for all x if and only if a ≤ 5. The largest value of 'a' that satisfies this condition is a = 5.