The integral values of k for which the equation (k - 2) x^2 + 8x + k + 4 = 0 has both the roots real, distinct and negative is :

For the equation (k - 2)x^2 + 8x + k + 4 = 0 to have real, distinct, and negative roots, the discriminant must be strictly positive. Calculating the discriminant gives 64 - 4(k - 2)(k + 4) > 0, which simplifies to 64 - 4(k^2 + 2k - 8) > 0 and further to k^2 + 2k - 24 < 0, restricting k to the range -6 < k < 4. Additionally, the sum of the roots (-8 / (k - 2)) and the product ((k + 4) / (k - 2)) must both be negative, which requires the leading coefficient to be positive, so k > 2. Testing integers in the range 2 < k < 4, only k = 3 satisfies all conditions.

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