The expression (4n-5)(5n-4) is equivalent to which of the following?

Options

A. `20n^2-41n+20`
B. `20n^2-39n+9`
C. `9n^2+41n+20`
D. `4n^2-18n+9`

Answer

To solve the expression (4n - 5)(5n - 4), we will use the distributive property (also known as the FOIL method for binomials). Here’s a step-by-step breakdown:

### Step 1: Identify the terms
We have two binomials:
- The first binomial is \(4n - 5\)
- The second binomial is \(5n - 4\)

### Step 2: Apply the distributive property (FOIL)
We will multiply each term in the first binomial by each term in the second binomial:
1. **First**: Multiply the first terms of each binomial:
\[ 4n \cdot 5n = 20n^2 \]

2. **Outer**: Multiply the outer terms:
\[ 4n \cdot (-4) = -16n \]

3. **Inner**: Multiply the inner terms:
\[ -5 \cdot 5n = -25n \]

4. **Last**: Multiply the last terms of each binomial:
\[ -5 \cdot (-4) = 20 \]

### Step 3: Combine all the results
Now, we combine all the results from the previous steps:
\[ 20n^2 - 16n - 25n + 20 \]

### Step 4: Simplify the expression
Combine the like terms (-16n and -25n):
\[ 20n^2 - 41n + 20 \]

### Final Result
The expression \((4n - 5)(5n - 4)\) simplifies to:
\[ 20n^2 - 41n + 20 \]

### Conclusion
Thus, the expression \((4n - 5)(5n - 4)\) is equivalent to \(20n^2 - 41n + 20\).

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