For all x, `(x + 4)(x-5) = ?`

Options

A. `x^(2) -20`
B. `x^(2) - x - 20`
C. `2x - 1`
D. `2x^(2) - 1`

Answer

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

### Step 1: Write down the expression
We start with the expression:
\[ (x + 4)(x - 5) \]

### Step 2: Apply the distributive property
We will distribute each term in the first binomial \((x + 4)\) to each term in the second binomial \((x - 5)\).

1. Multiply \(x\) by \(x\):
\[ x \cdot x = x^2 \]

2. Multiply \(x\) by \(-5\):
\[ x \cdot (-5) = -5x \]

3. Multiply \(4\) by \(x\):
\[ 4 \cdot x = 4x \]

4. Multiply \(4\) by \(-5\):
\[ 4 \cdot (-5) = -20 \]

### Step 3: Combine all the results
Now we combine all the results from the multiplications:
\[ x^2 - 5x + 4x - 20 \]

### Step 4: Simplify the expression
Next, we combine like terms:
\[ -5x + 4x = -1x \quad \text{or} \quad -x \]
Thus, the expression simplifies to:
\[ x^2 - x - 20 \]

### Final Result
The final result of the expression \((x + 4)(x - 5)\) is:
\[ x^2 - x - 20 \]

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