$$ \left. \begin{array} { l } { ( m + 9 ) ( m - 3 ) } \ { m ^ { 2 } - 12 m + 27 } \ { m ^ { 2 } + 6 m - 27 } \ { m ^ { 2 } - 6 m + 27 } \ { m ^ { 2 } + 12 m - 27 } \end{array} ight. $$

Question

$$ \left. \begin{array} { l } { ( m + 9 ) ( m - 3 ) } \ { m ^ { 2 } - 12 m + 27 } \ { m ^ { 2 } + 6 m - 27 } \ { m ^ { 2 } - 6 m + 27 } \ { m ^ { 2 } + 12 m - 27 } \end{array} ight. $$

UpStudy AI · Accepted Answer

It looks like you've provided a list of algebraic expressions involving the variable $$ m $$. Let's analyze each of them to understand their relationships and determine any equivalences.

### Given Expressions:
1. **$$ (m + 9)(m - 3) $$**
2. **$$ m^2 - 12m + 27 $$**
3. **$$ m^2 + 6m - 27 $$**
4. **$$ m^2 - 6m + 27 $$**
5. **$$ m^2 + 12m - 27 $$**

### Step-by-Step Analysis:

1. **Expand the First Expression:**
   
   Let's start by expanding the first expression to see which of the other expressions it matches.

$$
   (m + 9)(m - 3) = m \cdot m + m \cdot (-3) + 9 \cdot m + 9 \cdot (-3) = m^2 - 3m + 9m - 27 = m^2 + 6m - 27
   $$
   
   **Result:** $$ m^2 + 6m - 27 $$ (This matches **Expression 3**)

2. **Compare Other Expressions:**

- **Expression 2:** $$ m^2 - 12m + 27 $$
     - Not equivalent to $$ m^2 + 6m - 27 $$
   
   - **Expression 3:** $$ m^2 + 6m - 27 $$
     - **Equivalent to the expanded first expression**
   
   - **Expression 4:** $$ m^2 - 6m + 27 $$
     - Not equivalent to $$ m^2 + 6m - 27 $$
   
   - **Expression 5:** $$ m^2 + 12m - 27 $$
     - Not equivalent to $$ m^2 + 6m - 27 $$

### Conclusion:

- **Equivalent Expressions:**
  - **Expression 1:** $$ (m + 9)(m - 3) $$
  - **Expression 3:** $$ m^2 + 6m - 27 $$

These two expressions are equivalent because expanding $$ (m + 9)(m - 3) $$ results in $$ m^2 + 6m - 27 $$.

- **Non-Equivalent Expressions:**
  - **Expression 2:** $$ m^2 - 12m + 27 $$
  - **Expression 4:** $$ m^2 - 6m + 27 $$
  - **Expression 5:** $$ m^2 + 12m - 27 $$

These expressions do not match the expanded form of $$ (m + 9)(m - 3) $$ and are thus not equivalent to it.

### Additional Insights:

If you're looking to factor or simplify these expressions further, here's a brief overview:

- **Expression 2:** $$ m^2 - 12m + 27 $$
  - Can be factored as $$ (m - 3)(m - 9) $$ since $$ 3 \times 9 = 27 $$ and $$ 3 + 9 = 12 $$.

- **Expressions 4 & 5:** These do not factor neatly with integer coefficients, so they remain as they are unless complex or fractional factors are considered.

Feel free to provide more context or ask further questions if you need assistance with a specific aspect of these expressions!
$$ (m + 9)(m - 3) $$ is equivalent to $$ m^2 + 6m - 27 $$.

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