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GATE CS 2022 Q3 — General Aptitude

Let $r$ be a root of the equation $x^{2} + 2x + 6 = 0.$

Then the value of the expression $(r+2) (r+3) (r+4) (r+5)$ is

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Answer: D

Since $r$ is a root of the equation $x^2 + 2x + 6 = 0$ , we have:

r^2 + 2r + 6 = 0 \implies r^2 + 2r = -6 \quad \text{and} \quad r^2 = -2r - 6

We need to find the value of the expression:

E = (r+2)(r+3)(r+4)(r+5)

Grouping the terms as $(r+2)(r+5)$ and $(r+3)(r+4)$ , we get:

E = \left(r^2 + 7r + 10\right)\left(r^2 + 7r + 12\right)

Substituting $r^2 = -2r - 6$ into each term:

r^2 + 7r + 10 = (-2r - 6) + 7r + 10 = 5r + 4

r^2 + 7r + 12 = (-2r - 6) + 7r + 12 = 5r + 6

Now, multiply these simplified terms:

E = (5r + 4)(5r + 6) = 25r^2 + 50r + 24

E = 25(r^2 + 2r) + 24

Substituting $r^2 + 2r = -6$ into the equation:

E = 25(-6) + 24 = -150 + 24 = -126

Thus, the correct option is D.

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Published 10 Oct 2026

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