Question 1

Consider the polynomial $P(x)$ :

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

It is given that $f(x) \neq g(x)$ for all $x \in \mathbb{R}$, which means $P(x)$ has no real roots.

Since any polynomial of odd degree with real coefficients must have at least one real root, the degree of $P(x)$ cannot be 3 or 1. Thus, the coefficient of $x^3$ must be zero, implying $a_3 - b_3 = 0 \implies a_3 = b_3$.

Now, find the coefficient of $x^3$ in $h(x) = f(x + 1) - g(x + 2)$:

Coefficient of $x^3$ in $f(x + 1) = a_3 \cdot \binom{3}{3} + 1 \cdot \binom{4}{3} = a_3 + 4$.

Coefficient of $x^3$ in $g(x + 2) = b_3 \cdot \binom{3}{3} + 1 \cdot \binom{4}{3} \cdot 2^1 = b_3 + 8$.

Coefficient of $x^3$ in $h(x) = (a_3 + 4) - (b_3 + 8) = (a_3 - b_3) - 4$.

Since $a_3 = b_3$, the coefficient is $0 - 4 = -4$.