Which statement is true about this equation?

-4(2*p* + 5) + 8*p* = -11

**A.**

The equation has one solution, *p* = 2.

**B.**

The equation has one solution, *p* = -2.

**C.**

The equation has no solution.

**D.**

The equation has infinitely many solutions.

To determine which statement is true about the equation

\[
-4(2p + 5) + 8p = -11,
\]

let's solve it step by step.

### Step 1: Distribute the -4
Distribute \(-4\) to both terms inside the parentheses:

\[
-4 \cdot 2p + (-4) \cdot 5 = -8p - 20
\]

So, the equation becomes:

\[
-8p - 20 + 8p = -11
\]

### Step 2: Combine like terms
Combine the like terms on the left-hand side:

\[
-8p + 8p - 20 = -11
\]

Since \(-8p + 8p\) cancels out, you're left with:

\[
-20 = -11
\]

### Step 3: Analyze the result
The equation simplifies to \(-20 = -11\), which is a contradiction. This means that the equation is never true, regardless of the value of \(p\).

### Conclusion:
The equation has no solution.

So, the correct answer is **C.** The equation has no solution.

Would you like any further details or explanations? Here are some additional questions to consider:

1. How do we identify when an equation has no solution?
2. What would happen if the equation resulted in a true statement, like \(0 = 0\)?
3. How can you verify your solution to a linear equation?
4. What is the significance of combining like terms when solving equations?
5. How does distributing a negative sign affect the terms inside the parentheses?

**Tip:** Always check for contradictions or identities when solving equations to determine if there are no solutions or infinitely many solutions.

Explore the solution process for the equation -4(2p + 5) + 8p = -11. Understand the application of the distributive property, combining like terms, and analyzing contradictions in linear equations. Suitable for Grades 7-9 students studying algebra.

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