1. Problem: Find the natural number $n$ such that the equality $\left(1 + \frac{z}{3} x\right)^n \cdot (3 + n x)^2 = 9 + 84 x + \ldots$ holds. 2. We want to expand the left side and match the constant and linear terms with the right side. 3. First, expand $(3 + n x)^2$ using the binomial formula: $$ (3 + n x)^2 = 9 + 2 \cdot 3 \cdot n x + (n x)^2 = 9 + 6 n x + n^2 x^2 $$ 4. Next, expand $\left(1 + \frac{z}{3} x\right)^n$ using the binomial theorem up to the linear term: $$ \left(1 + \frac{z}{3} x\right)^n = 1 + n \cdot \frac{z}{3} x + \ldots $$ 5. Multiply the expansions up to the linear term: $$ \left(1 + n \frac{z}{3} x\right) \cdot (9 + 6 n x) = 9 \cdot 1 + 9 \cdot n \frac{z}{3} x + 6 n x \cdot 1 + \ldots = 9 + \left(9 n \frac{z}{3} + 6 n\right) x + \ldots $$ 6. Simplify the coefficient of $x$: $$ 9 n \frac{z}{3} + 6 n = 3 n z + 6 n = n (3 z + 6) $$ 7. The right side is $9 + 84 x + \ldots$, so equate the constant and linear terms: - Constant term: $9 = 9$ (matches) - Linear term: $n (3 z + 6) = 84$ 8. Solve for $n$: $$ n = \frac{84}{3 z + 6} $$ 9. Since $n$ is a natural number, $3 z + 6$ must divide 84 exactly. 10. Without a given value for $z$, we cannot find a unique $n$. If $z$ is known, plug it in and compute $n$. Final answer: $$ n = \frac{84}{3 z + 6} $$ where $n$ is a natural number and $z$ is a parameter from the problem.