To determine the correct expansion of [tex]\((3x - 1)(4x^2 + 5)\)[/tex], we need to use the distributive property, also known as the FOIL method for binomials, to expand the expression.
Here’s the step-by-step process:
1. Distribute [tex]\(3x\)[/tex] to each term inside the second parenthesis:
- [tex]\(3x \cdot 4x^2 = 12x^3\)[/tex]
- [tex]\(3x \cdot 5 = 15x\)[/tex]
2. Distribute [tex]\(-1\)[/tex] to each term inside the second parenthesis:
- [tex]\(-1 \cdot 4x^2 = -4x^2\)[/tex]
- [tex]\(-1 \cdot 5 = -5\)[/tex]
Now, combining all these products together:
[tex]\[ 12x^3 + 15x - 4x^2 - 5 \][/tex]
From the given options, we can compare our calculated steps to find the matching expansion:
- Option A: [tex]\(3x \cdot 4x^2 + (-1) \cdot 4x^2 + 4x^2 \cdot 5 + (-1) \cdot 5\)[/tex]
- This includes terms that are not correctly distributed from the original expression and is incorrect.
- Option B: [tex]\(3x \cdot 4x^2 + 3x \cdot 5 + (-1) \cdot 4x^2 + (-1) \cdot 5\)[/tex]
- This correctly matches our calculated distribution steps.
- Option C: [tex]\(3x \cdot 4x^2 + 3x \cdot 5 + 1 \cdot 4x^2 + 1 \cdot 5\)[/tex]
- This also includes incorrect signs and is not the correct distribution.
Therefore, the correct expansion given among the options is:
[tex]\[ \boxed{\text{Option B}\: 3x \cdot 4x^2 + 3x \cdot 5 + (-1) \cdot 4x^2 + (-1) \cdot 5} \][/tex]
