JEE Advance 2025 Paper 1

Let $\mathbb{R}$   denote the set of all real numbers. Let $a_{i}, b_{i} \in \mathbb{R}$   for $i \in \{1, 2, 3\}$  . Define the functions $f: \mathbb{R} \to \mathbb{R}$  , $g: \mathbb{R} \to \mathbb{R}$  , and $h: \mathbb{R} \to \mathbb{R}$   by $f(x) = a_{1} + 10x + a_{2}x^{2} + a_{3}x^{3} + x^{4}$  , $g(x) = b_{1} + 3x + b_{2}x^{2} + b_{3}x^{3} + x^{4}$  , and $h(x) = f(x + 1) - g(x + 2)$  . If $f(x) \neq g(x)$   for every $x \in \mathbb{R}$  , then the coefficient of $x^{3}$   in $h(x)$   is