To find the product of the binomials [tex]\( (2x - 1) \)[/tex] and [tex]\( (x + 4) \)[/tex], we can use the distributive property (also known as the FOIL method for binomials). Here's a detailed, step-by-step solution:
1. Distribute each term in the first binomial to each term in the second binomial.
[tex]\[
(2x - 1)(x + 4)
\][/tex]
2. First, multiply the first terms of each binomial:
[tex]\[
2x \cdot x = 2x^2
\][/tex]
3. Outer, multiply the outer terms of the binomials:
[tex]\[
2x \cdot 4 = 8x
\][/tex]
4. Inner, multiply the inner terms of the binomials:
[tex]\[
-1 \cdot x = -x
\][/tex]
5. Last, multiply the last terms of each binomial:
[tex]\[
-1 \cdot 4 = -4
\][/tex]
6. Combine all the terms together:
[tex]\[
2x^2 + 8x - x - 4
\][/tex]
7. Simplify by combining like terms:
[tex]\[
2x^2 + 7x - 4
\][/tex]
Hence, the product of [tex]\( (2x-1)(x+4) \)[/tex] is:
[tex]\[
2x^2 + 7x - 4
\][/tex]
Among the given options, the correct one is:
[tex]\[
2x^2 + 7x - 4
\][/tex]
