1. **State the problem:** We want to find the value of $n$ such that the polynomial $x^3 + 22x + n$ becomes a perfect square trinomial. 2. **Understand the form of a perfect square trinomial:** A perfect square trinomial is generally of the form $\left(ax^m + b\right)^2 = a^2x^{2m} + 2abx^m + b^2$. 3. **Analyze the given polynomial:** The polynomial is $x^3 + 22x + n$. Notice the highest power is 3, which is odd, but perfect square trinomials have even powers because of squaring. 4. **Check if the polynomial can be a perfect square:** Since the highest power is 3, it cannot be expressed as the square of a binomial with integer or rational coefficients because squaring a binomial results in even powers. 5. **Conclusion:** There is no value of $n$ that makes $x^3 + 22x + n$ a perfect square trinomial. Therefore, the polynomial cannot be factored as the square of a binomial for any $n$.