Let \(\mathbb{R}\) denote the set of all real numbers. Let \({a}_{i},{b}_{i}\in \mathrm{ℝ}\) for \(i\in {1,2,3}\). Defin…

Let \(\mathbb{R}\) denote the set of all real numbers. Let \({a}_{i},{b}_{i}\in \mathrm{ℝ}\) for \(i\in {1,2,3}\).
Define the functions \(f:\mathrm{ℝ}\to \mathrm{ℝ},g:\mathrm{ℝ}\to \mathrm{ℝ}\), and \(h:\mathrm{ℝ}\to \mathrm{ℝ}\) by

\(f(x)={a}_{1}+10x+{a}_{2}{x}^{2}+{a}_{3}{x}^{3}+{x}^{4}\\ g(x)={b}_{1}+3x+{b}_{2}{x}^{2}+{b}_{3}{x}^{3}+{x}^{4}\\ h(x)=f(x+1)-g(x+2)\)

If \(f(x) \neq g(x)\) for every \(x \in \mathbb{R}\), then the coefficient of \(x^3\) in \(h(x)\) is

[JEE Advanced 2025]