To simplify the given expression, we will expand each term and then combine like terms.

Expanding the first term:
(а-2)(а^2+2а+4)
= a(a^2) + 2a(a) + 4a - 2a^2 - 4
= a^3 + 2a^2 + 4a - 2a^2 - 4
= a^3 + 4a - 4

Expanding the second term:
(а+с)(а^2-ас+с^2)
= a(a^2) - a(ас) + a(c^2) + c(a^2) - c(ас) + c(c^2)
= a^3 - a^2c + ac^2 + a^2c - ac^2 + c^3
= a^3 + c^3

Expanding the third term:
(c+2)(c^2-2c+4)
= c(c^2) - 2c(c) + 4c + 2(c^2) - 4c + 8
= c^3 - 2c^2 + 4c + 2c^2 - 4c + 8
= c^3 + 8

Now substituting these results back into the original equation:
(a^3 + 4a - 4) - (a^3 + c^3) + (c^3 + 8) = 0
a^3 + 4a - 4 - a^3 - c^3 + c^3 + 8 = 0
4a - 4 + 8 = 0
4a + 4 = 0
4a = -4
a = -1

Therefore, the solution to the given equation is a = -1.