Answer :

To express the product of the binomials [tex]\((3x + 1)(x - 4)\)[/tex] as a trinomial, follow these steps:

1. Identify the terms to be multiplied:

The two binomials are [tex]\((3x + 1)\)[/tex] and [tex]\((x - 4)\)[/tex].

2. Apply the distributive property (also known as the FOIL method for binomials):

Multiply each term in the first binomial by each term in the second binomial.
[tex]\[ (3x + 1)(x - 4) = 3x \cdot x + 3x \cdot (-4) + 1 \cdot x + 1 \cdot (-4) \][/tex]

3. Perform the individual multiplications:

- [tex]\(3x \cdot x = 3x^2\)[/tex]
- [tex]\(3x \cdot (-4) = -12x\)[/tex]
- [tex]\(1 \cdot x = x\)[/tex]
- [tex]\(1 \cdot (-4) = -4\)[/tex]

4. Combine the like terms:

The expression obtained after multiplication is:
[tex]\[ 3x^2 - 12x + x - 4 \][/tex]

Combine the [tex]\(x\)[/tex] terms:
[tex]\[ -12x + x = -11x \][/tex]

5. Write the final trinomial:

The trinomial that represents the product of [tex]\((3x + 1)(x - 4)\)[/tex] is:
[tex]\[ 3x^2 - 11x - 4 \][/tex]

Therefore, expressing [tex]\((3x + 1)(x - 4)\)[/tex] as a trinomial gives:
[tex]\[ 3x^2 - 11x - 4 \][/tex]