Let's simplify the given expression step by step:

$(2m + \frac{1}{4}) \cdot \left(-\frac{1}{2}m + 4\right)$

Step 1: Distribute each term

Using the distributive property:

$(2m + \frac{1}{4}) \cdot \left(-\frac{1}{2}m + 4\right) = 2m \cdot \left(-\frac{1}{2}m\right) + 2m \cdot 4 + \frac{1}{4} \cdot \left(-\frac{1}{2}m\right) + \frac{1}{4} \cdot 4$

Step 2: Multiply each term

• $2m \cdot (-\frac{1}{2}m) = -m^2$
• $2m \cdot 4 = 8m$
• $\frac{1}{4} \cdot (-\frac{1}{2}m) = -\frac{1}{8}m$
• $\frac{1}{4} \cdot 4 = 1$

Step 3: Combine like terms

$-m^2 + 8m - \frac{1}{8}m + 1$

Since $8m - \frac{1}{8}m = \frac{64}{8}m - \frac{1}{8}m = \frac{63}{8}m$, we rewrite:

$-m^2 + \frac{63}{8}m + 1$

Final Answer:

$-m^2 + \frac{63}{8}m + 1$

Would you like a more detailed breakdown or another example?

Related Questions:

1. How do you expand and simplify algebraic expressions?
2. How does the distributive property work in algebra?
3. What are like terms, and how do you combine them?
4. How do you multiply fractions with variables?
5. How can factoring help simplify expressions like this?

Tip:

Always watch out for negative signs when distributing—it's a common place for mistakes!