Find maximum value of `(6-x) (x+4)`

Answer

To find the maximum value of the expression \( (6 - x)(x + 4) \), we can follow these steps:

### Step 1: Expand the expression
First, we will expand the expression \( (6 - x)(x + 4) \).

\[ (6 - x)(x + 4) = 6x + 24 - x^2 - 4x = -x^2 + 2x + 24 \]

### Step 2: Differentiate the expression
Next, we will differentiate the expanded expression with respect to \( x \).

\[ \frac{d}{dx}(-x^2 + 2x + 24) = -2x + 2 \]

### Step 3: Set the derivative to zero
To find the critical points, we set the derivative equal to zero.

\[ -2x + 2 = 0 \]

### Step 4: Solve for \( x \)
Now, we solve for \( x \).

\[ -2x = -2 \implies x = 1 \]

### Step 5: Find the maximum value
Now, we will substitute \( x = 1 \) back into the original expression to find the maximum value.

\[ (6 - 1)(1 + 4) = 5 \times 5 = 25 \]

### Conclusion
Thus, the maximum value of the expression \( (6 - x)(x + 4) \) is \( \boxed{25} \).

---