JEE Advanced 2025 · Paper 1, Mathematics question 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

$$\begin{aligned} f(x) &= a_1 + 10x + a_2x^2 + a_3x^3 + x^4,\\ g(x) &= b_1 + 3x + b_2x^2 + b_3x^3 + x^4,\\ h(x) &= f(x+1) - g(x+2). \end{aligned}$$

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