Find the product of the polynomials (2x+1)(x−4)

To find the product of the polynomials \((2x + 1)(x - 4)\), we'll use the distributive property (also known as the FOIL method for binomials).

1. **First:** Multiply the first terms:
\[
2x \cdot x = 2x^2
\]

2. **Outer:** Multiply the outer terms:
\[
2x \cdot (-4) = -8x
\]

3. **Inner:** Multiply the inner terms:
\[
1 \cdot x = x
\]

4. **Last:** Multiply the last terms:
\[
1 \cdot (-4) = -4
\]

Now, combine all the results:
\[
2x^2 - 8x + x - 4
\]

Combine like terms:
\[
2x^2 + (-8x + x) - 4 = 2x^2 - 7x - 4
\]

Thus, the product of the polynomials \((2x + 1)(x - 4)\) is:
\[
\boxed{2x^2 - 7x - 4}
\]