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 will multiply each term in the first polynomial by each term in the second polynomial:
- First: \( 2x \cdot x = 2x^2 \)
- Outer: \( 2x \cdot (-4) = -8x \)
- Inner: \( 1 \cdot x = 1x \)
- Last: \( 1 \cdot (-4) = -4 \)
Now combine all these results:
\[ 2x^2 - 8x + 1x - 4 \]
Combine the like terms (-8x and 1x):
\[ 2x^2 - 7x - 4 \]
Thus, the product of the polynomials \( (2x + 1)(x - 4) \) is
\[ \boxed{2x^2 - 7x - 4} \]