In this article, we investigate the growth of meromorphic solutions of
a(z)(△cηη)2+(b2(z)η2(z)+b1(z)η(z)+b0(z))△cηη=d4(z)η4(z)+d3(z)η3(z)+d2(z)η2(z)+d1(z)η(z)+d0(z),
where a(z), b_i(z) for i = 0, 1, 2 and d_j(z) for j = 0, ..., 4 are given functions, △_cη = η(z + c) − η(z) with c ∈ ℂ\{0}. In particular, when the a(z), the b_i(z) and the d_j(z) are polynomials, and d₄(z) ≡ 0, we shall show that if η(z) is a transcendental entire solution of finite order, and either deg a(z) ̸= deg d₀(z)+1, or, deg a(z) = deg d₀(z)+1 and ρ(η)≠12, then ρ(η) ≥ 1.

Keywords: Entire functions, Difference equations, Value distribution, Finite order