Let r be a root of the equation 𝑥^2 + 2𝑥 + 6 = 0. Then the value of the…

We need the value of (r + 2)(r + 3)(r + 4)(r + 5) where r satisfies r^2 + 2r + 6 = 0.

• Use the equation to express r^2 in terms of r: r^2 = -2r - 6.

• Pair the factors: (r + 2)(r + 5) = r^2 + 7r + 10 = (−2r − 6) + 7r + 10 = 5r + 4.

• Similarly, (r + 3)(r + 4) = r^2 + 7r + 12 = (−2r − 6) + 7r + 12 = 5r + 6.

• Multiply the two results: (5r + 4)(5r + 6) = 25r^2 + 50r + 24. Substitute r^2 = −2r − 6:

• 25(−2r − 6) + 50r + 24 = −50r − 150 + 50r + 24 = −126.

Final answer: -126