To determine which of the provided options is equivalent to the expression [tex]\((4z + 3)(z - 2)\)[/tex], we will expand this expression step by step.
1. First, apply the distributive property (also known as FOIL method for binomials) to the expression [tex]\((4z + 3)(z - 2)\)[/tex]:
[tex]\[(4z + 3)(z - 2) = (4z)(z) + (4z)(-2) + (3)(z) + (3)(-2)\][/tex]
2. Calculate each term:
[tex]\[
\begin{align*}
(4z)(z) & = 4z^2 \\
(4z)(-2) & = -8z \\
(3)(z) & = 3z \\
(3)(-2) & = -6
\end{align*}
\][/tex]
3. Combine the calculated terms:
[tex]\[
4z^2 + (-8z) + (3z) + (-6)
\][/tex]
4. Simplify the expression by combining like terms:
[tex]\[
4z^2 - 8z + 3z - 6 = 4z^2 - 5z - 6
\][/tex]
5. So the expanded form of the expression [tex]\((4z + 3)(z - 2)\)[/tex] is [tex]\(4z^2 - 5z - 6\)[/tex].
Now, let's match this result with the given options:
[tex]\[
\begin{align*}
1. & \ 4z^2 - 5z - 6 \\
2. & \ 4z^2 - 6 \\
3. & \ 4z^2 - 5 \\
4. & \ 4z^2 + 5z - 6 \\
5. & \ 4z^2 - 3z - 5
\end{align*}
\][/tex]
The expanded form ( [tex]\(4z^2 - 5z - 6\)[/tex] ) matches exactly with the first option.
Therefore, the expression [tex]\((4z + 3)(z - 2)\)[/tex] is equivalent to [tex]\(4z^2 - 5z - 6\)[/tex].
