To find the product of the polynomials \( (2x + 1)(x - 4) \), we will use the distributive property (also known as the FOIL method for binomials).
- Multiply the first terms: \( 2x \cdot x = 2x^2 \)
- Multiply the outer terms: \( 2x \cdot (-4) = -8x \)
- Multiply the inner terms: \( 1 \cdot x = 1x \) (or just \( x \))
- Multiply the last terms: \( 1 \cdot (-4) = -4 \)
Now, we combine all these results: \[ 2x^2 - 8x + x - 4 \]
Combine the like terms \( -8x + x = -7x \):
Thus, the final polynomial is: \[ 2x^2 - 7x - 4 \]
So, filling in the blanks: \[ __x^2 + __x + __ \] becomes \[ 2x^2 + (-7)x + (-4) \]
Therefore, the answer is: \[ 2x^2 - 7x - 4 \]