To find the product of the polynomials \( (2x + 1)(x - 4) \), we can use the distributive property (also known as the FOIL method for binomials).
We perform the multiplication as follows:
- First: Multiply the first terms \( 2x \cdot x = 2x^2 \).
- Outer: Multiply the outer terms \( 2x \cdot (-4) = -8x \).
- Inner: Multiply the inner terms \( 1 \cdot x = 1x \).
- Last: Multiply the last terms \( 1 \cdot (-4) = -4 \).
Now, we combine all these results:
\[ 2x^2 - 8x + 1x - 4 \]
Next, we combine like terms (\(-8x\) and \(1x\)):
\[ 2x^2 - 7x - 4 \]
Thus, the product of the polynomials \( (2x + 1)(x - 4) \) is
\[ \boxed{2x^2 - 7x - 4} \]