You're correct in noting the need for clear differentiation. Let's clarify the polynomial $p(x) = (x^2 + x + 1)(x + 1)x$:

1. Monomial factor: A monomial is a single term, which in this case is $x$. The term $x$ stands alone as a monomial factor.

2. Binomial factor: A binomial has two terms. In the given polynomial, the factor $(x + 1)$ is a binomial.

3. Trinomial factor: A trinomial has three terms. In this polynomial, the factor $(x^2 + x + 1)$ is a trinomial.

Hence, let's reassess the provided options:

• The monomial factor of $p(x)$ is $5$: This is incorrect.
• The monomial factor of $p(x)$ is $x$: This is correct. The factor $x$ is a monomial.
• The binomial factor of $p(x)$ is $(x + 1)$: This is correct. The factor $(x + 1)$ is a binomial.
• The binomial factor of $p(x)$ is $(x^2 + x + 1)$: This is incorrect. $(x^2 + x + 1)$ is a trinomial.
• The trinomial factor of $p(x)$ is $(x^2 + x + 1)$: This is correct. $(x^2 + x + 1)$ is a trinomial.

Thus, the correct set of options are:

• The monomial factor of $p(x)$ is $x$.
• The binomial factor of $p(x)$ is $(x + 1)$.
• The trinomial factor of $p(x)$ is $(x^2 + x + 1)$.

Would you like further details or have any other questions?

Here are five questions you might consider next:

1. How do you factorize a polynomial?
2. What are the definitions of monomials, binomials, and trinomials?
3. Can you explain the process of polynomial multiplication?
4. How do you identify the degree of a polynomial?
5. What are some common techniques for solving polynomial equations?

Tip: When identifying factors, break the polynomial into its simplest components and count the number of terms in each factor to determine if it is a monomial, binomial, or trinomial.