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 $f : \mathbb{R} \rightarrow \mathbb{R}, g : \mathbb{R} \rightarrow \mathbb{R}$; and $h : \mathbb{R} \rightarrow \mathbb{R}$ by $f(x) = a_{1} + 10x + a_{2}x^{2} + a_{3}x^{3} + x^{4},$ $g(x) = b_{1} + 3x + b_{2}x^{2} + b_{3}x^{3} + x^{4},$ $h(x) = f(x+1) - g(x+2).$ If $f(x) \neq g(x)$ for every $x \in \mathbb{R}$, then the coefficient of $x^{3}$ in $h(x)$ is

$f(x) = a_1 + 10x + a_2 x^2 + a_3 x^3 + x^4$

$g(x) = b_1 + 3x + b_2 x^2 + b_3 x^3 + x^4$

$h(x) = f(x+1) - g(x+2)$

$f(x) \neq g(x) \quad \forall\, x \in \mathbb{R}$

$\text{Find: coefficient of } x^3 \text{ in } h(x)$

$\text{1. Use } f \neq g \text{ to relate } a_3, b_3$

$\text{2. Find the coefficient of } x^3 \text{ in } h(x)$

$f(x) - g(x) = (a_1 - b_1) + 7x + (a_2 - b_2)x^2 + (a_3 - b_3)x^3$

$f(x) \neq g(x) \quad \forall\, x \in \mathbb{R}$

$\iff f(x) - g(x) \neq 0 \quad \forall\, x \in \mathbb{R}$

$\Rightarrow f - g \text{ has no real root}$

$\deg(f-g) \le 3$

$\text{odd degree} \Rightarrow \text{at least one real root}$

$\therefore a_3 - b_3 = 0 \Rightarrow a_3 = b_3$

$h(x) = f(x+1) - g(x+2)$

$a_3(x+1)^3 \to a_3 x^3$

$(x+1)^4 = x^4 + 4x^3 + 6x^2 + 4x + 1$

$\text{coeff of } x^3 \text{ in } f(x+1) = a_3 + 4$

$b_3(x+2)^3 \to b_3 x^3$

$(x+2)^4 = x^4 + 8x^3 + 24x^2 + 32x + 16$

$\text{coeff of } x^3 \text{ in } g(x+2) = b_3 + 8$

$\text{coeff in } h = (a_3 + 4) - (b_3 + 8)$

$= (a_3 - b_3) - 4$

$= (a_3 - b_3) - 4 = 0 - 4$

$\boxed{\text{Option (3) } -4}$