Question: Select the expressions that are equivalent to $4(-4z - 6) + 4z$.
- $-6(-4z + 4) + 4z$
- $-4(4z - 6) + 4z$
- $-12z - 24$
- $2(-8z - 12) + 4z$
1. **State the problem:** We need to find which expressions are equivalent to $$4(-4z - 6) + 4z$$.
2. **Simplify the original expression:**
$$4(-4z - 6) + 4z = 4 \times -4z + 4 \times -6 + 4z = -16z - 24 + 4z$$
Combine like terms:
$$-16z + 4z - 24 = -12z - 24$$
3. **Check each option for equivalence:**
- Option 1: $$-6(-4z + 4) + 4z = -6 \times -4z + -6 \times 4 + 4z = 24z - 24 + 4z = 28z - 24$$
Not equivalent to $$-12z - 24$$.
- Option 2: $$-4(4z - 6) + 4z = -4 \times 4z + -4 \times -6 + 4z = -16z + 24 + 4z$$
Combine like terms:
$$-16z + 4z + 24 = -12z + 24$$
Not equivalent to $$-12z - 24$$.
- Option 3: $$-12z - 24$$
This matches the simplified original expression exactly.
- Option 4: $$2(-8z - 12) + 4z = 2 \times -8z + 2 \times -12 + 4z = -16z - 24 + 4z$$
Combine like terms:
$$-16z + 4z - 24 = -12z - 24$$
Equivalent to the original expression.
4. **Final answer:** The expressions equivalent to $$4(-4z - 6) + 4z$$ are:
- $$-12z - 24$$
- $$2(-8z - 12) + 4z$$