Write as a single fraction in its simplest form 2/m+2 -6(m-1)/m square -4

To simplify the expression \( \frac{2}{m+2} - \frac{6(m-1)}{m^2} - 4 \) as a single fraction, follow these steps:

### Step 1: Express \( -4 \) as a fraction
First, rewrite the constant \(-4\) as a fraction with the same denominator as the rest:
\[
-4 = \frac{-4m^2(m+2)}{m^2(m+2)} = \frac{-4m^2(m+2)}{m^2(m+2)}
\]

### Step 2: Find the common denominator
The denominators of the fractions are \( m+2 \), \( m^2 \), and \( m^2(m+2) \). The least common denominator (LCD) will be:
\[
\text{LCD} = m^2(m+2)
\]

### Step 3: Rewrite each fraction with the LCD
Now, rewrite each term with the common denominator \( m^2(m+2) \):
\[
\frac{2}{m+2} = \frac{2m^2}{m^2(m+2)}
\]
\[
\frac{-6(m-1)}{m^2} = \frac{-6(m-1)(m+2)}{m^2(m+2)}
\]
\[
-4 = \frac{-4m^2(m+2)}{m^2(m+2)}
\]

### Step 4: Combine the fractions
Now, combine all terms over the common denominator:
\[
\frac{2m^2}{m^2(m+2)} - \frac{6(m-1)(m+2)}{m^2(m+2)} - \frac{4m^2(m+2)}{m^2(m+2)} = \frac{2m^2 - 6(m-1)(m+2) - 4m^2(m+2)}{m^2(m+2)}
\]

### Step 5: Simplify the numerator
Let's simplify the numerator step by step:
1. Expand \( -6(m-1)(m+2) \):
\[
-6(m-1)(m+2) = -6[m^2 + 2m - m - 2] = -6(m^2 + m - 2) = -6m^2 - 6m + 12
\]
2. Expand \( -4m^2(m+2) \):
\[
-4m^2(m+2) = -4m^3 - 8m^2
\]
3. Combine all the terms:
\[
2m^2 - 6m^2 - 6m + 12 - 4m^3 - 8m^2
\]
Combine like terms:
\[
= -4m^3 - 12m^2 - 6m + 12
\]

### Step 6: Write the final simplified fraction
The simplified expression is:
\[
\frac{-4m^3 - 12m^2 - 6m + 12}{m^2(m+2)}
\]

This is the final answer.

Would you like further details, or do you have any questions?

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**Relative Questions:**
1. How would the process change if the expression involved division instead of subtraction?
2. What is the importance of finding a common denominator in this context?
3. How would you simplify the expression if \( m \) had a specific value?
4. Could there be any potential restrictions on the value of \( m \) in this fraction?
5. How would you approach combining like terms in more complex polynomials?

**Tip:** When simplifying fractions with polynomials, always check for common factors to reduce the expression to its simplest form.

Write as a single fraction in its simplest form 2/(m+2) - 6(m-1)/m^2 - 4

This guide provides a step-by-step solution to simplify the expression 2/(m+2) - 6(m-1)/m^2 - 4 into a single fraction. Key algebra concepts such as finding a common denominator and simplifying polynomials are explored. Suitable for students in Grades 9-11.

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