Let's analyze the problem step by step.

We are tasked with identifying expressions equivalent to $8(4x + 5) + 8x$.

Step 1: Simplify the given expression

Start by distributing $8$ to the terms inside the parentheses: $8(4x + 5) + 8x = 8 \cdot 4x + 8 \cdot 5 + 8x = 32x + 40 + 8x.$

Combine like terms ($32x + 8x$): $32x + 8x + 40 = 40x + 40.$

So the simplified form is $40x + 40$.

Step 2: Analyze the choices

Now, match each given option against $40x + 40$:

1. $40x + 40$: ✅ Correct, this matches the simplified form.
2. $(5 + 4x)8 + 8x$: ✅ Correct, rewriting this gives $8(4x + 5) + 8x$, which matches the original expression.
3. $4(8x + 10) + 8x$: ✅ Correct, simplifying $4(8x + 10)$ gives $32x + 40$, and adding $8x$ gives $40x + 40$.
4. $(4x + 5)8 + 8x$: ✅ Correct, this is another form of the original expression.

Final Answer:

All options are equivalent to $8(4x + 5) + 8x$.

Let me know if you want additional clarification or a breakdown!

Expansion Questions:

1. What are the steps for distributing and combining like terms?
2. How can equivalent expressions help solve equations?
3. What is the significance of factoring in simplifying expressions?
4. How can you check your work when simplifying linear expressions?
5. What are common mistakes to avoid when simplifying expressions?

Tip:

Always simplify expressions fully and compare all terms systematically to ensure equivalence.