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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:
\[
\boxed{2x^2 - 7x - 4}
\]