If x-2 is a factor of polynomial x+4x2-3x-18, then it's other factors are

Reading the polynomial as x^3 + 4x^2 - 3x - 18, put x = 2: 8 + 16 - 6 - 18 = 0, so x - 2 is a factor. Synthetic division with root 2 on the coefficients 1, 4, -3, -18 gives the quotient 1, 6, 9 with remainder 0. So the quotient is x^2 + 6x + 9, which is a perfect square, (x + 3)^2. The full factorisation is (x - 2)(x + 3)(x + 3), so the other factors are x + 3 and x + 3. Option C is correct.

• A: x + 2 would need the polynomial to vanish at x = -2. The value there is -8 + 16 + 6 - 18 = -4, so it is not a factor.
• B: x - 3 would need a root at x = 3. The value there is 27 + 36 - 9 - 18 = 36, not zero.
• D: x + 1 would need a root at x = -1. The value there is -1 + 4 + 3 - 18 = -12, not zero.

Core concept: the factor theorem, synthetic division, and spotting perfect squares. This is basic algebra, but the same tools come up in Mains answers on polynomial and quadratic structure.

Trap: aspirants find one factor and then pick two distinct factors from the options, choosing A or D. The repeated root (x + 3) is the point of the question.