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 $f \colon \mathbb{R} \to \mathbb{R}$  , $g \colon \mathbb{R} \to \mathbb{R}$  , and $h \colon \mathbb{R} \to \mathbb{R}$   by $f(x) = a_1 + 10x + a_2x^2 + a_3x^3 + x^4$  , $g(x) = b_1 + 3x + b_2x^2 + b_3x^3 + x^4$  , 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