https://file.aiquickdraw.com/mathbot/file/1725684522422m135obgu.jpg

The problem in the image asks to simplify the following expression:

\[
\frac{16 \cdot 6(2m - 1) - 2 \cdot 2m^2 + 4 \cdot m}{16 - 2 \cdot m^2 - 2 \cdot m}
\]

### Step-by-Step Solution:

1. **Distribute and simplify the numerator:**

   \[
   16 \cdot 6(2m - 1) = 96(2m - 1) = 192m - 96
   \]
   
   \[
   2 \cdot 2m^2 = 4m^2
   \]
   
   \[
   4 \cdot m = 4m
   \]
   
   So, the numerator becomes:
   
   \[
   192m - 96 - 4m^2 + 4m = -4m^2 + 196m - 96
   \]

2. **Simplify the denominator:**

   \[
   16 - 2m^2 - 2m
   \]

3. **Rewriting the expression:**

   Now, the expression becomes:

   \[
   \frac{-4m^2 + 196m - 96}{16 - 2m^2 - 2m}
   \]

4. **Factor both the numerator and denominator if possible:**

   Numerator:
   
   \[
   -4m^2 + 196m - 96
   \]
   
   The numerator can be factored by finding the roots using the quadratic formula, or by recognizing common factors if it's factorable.
   
   Denominator:
   
   \[
   16 - 2m^2 - 2m
   \]
   
   The denominator can also be factored similarly.

5. **Further Simplification:**
   
   If the numerator and denominator share a common factor, they can be further simplified. If not, this is the simplified expression.

Do you want further details on factoring or checking for common factors?

### Related Questions:
1. How do you apply the quadratic formula to solve \( ax^2 + bx + c = 0 \)?
2. What is the process of factoring quadratic expressions?
3. How do you identify and cancel out common factors in rational expressions?
4. How can the distributive property be applied to more complex expressions?
5. What are the special cases of factoring that can be applied to simplify expressions?

### Tip:
Always check for common factors in the numerator and denominator of a rational expression to simplify it to its lowest terms.

Simplify the following expression: (16 * 6(2m - 1) - 2 * 2m^2 + 4 * m) / (16 - 2 * m^2 - 2 * m)

Learn how to simplify rational expressions with quadratic equations by applying algebraic techniques like factoring, using the quadratic formula, and finding common factors. Suitable for students in Grades 9-12.

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