Two equal roots means a discriminant of zero.
(k + 2)x² + 4k = (4k + 2) x
(k + 2)x²- 2 (2k + 1) x + 4k = 0
When we have an even linear term it's easier to generate 1/4 of the usual discrimant,
0 = d/4 = (b/2)^2 - ac = (2k + 1)² - 4k(k+2) = 4k² + 4k + 1 - 4k² - 8k = -4k + 1
4k=1
k=1/4
Answer: k=1/4
Check:
(k + 2)x² + 4k = (4k + 2) x
(9/4)x² - 3x + 1 = 0
9x² - 12x + 4 = 0
(3x - 2)² = 0
A perfect square, two equal roots, good