`(x+2)(x-5)(x-6)(x+1)=144`:

Options

A. `-1, -2, -3`
B. `7, -3, 2`
C. `2, -3, 5`
D. none of these

Answer

To solve the equation \((x+2)(x-5)(x-6)(x+1) = 144\), we will follow these steps:

### Step 1: Expand the Left Side
First, we will expand the left-hand side of the equation. We can group the factors for easier multiplication:

\[ (x+2)(x+1) \quad \text{and} \quad (x-5)(x-6) \]

Calculating the first group:

\[ (x+2)(x+1) = x^2 + 3x + 2 \]

Calculating the second group:

\[ (x-5)(x-6) = x^2 - 11x + 30 \]

Now we multiply these two results together:

\[ (x^2 + 3x + 2)(x^2 - 11x + 30) \]

### Step 2: Multiply the Two Quadratics
Now we will expand this product:

\[ = x^4 - 11x^3 + 30x^2 + 3x^3 - 33x^2 + 90x + 2x^2 - 22x + 60 \]

Combining like terms:

\[ = x^4 - 8x^3 + (30 - 33 + 2)x^2 + (90 - 22)x + 60 \]
\[ = x^4 - 8x^3 - x^2 + 68x + 60 \]

### Step 3: Set the Equation to Zero
Now we set the equation equal to zero by moving 144 to the left side:

\[ x^4 - 8x^3 - x^2 + 68x + 60 - 144 = 0 \]
\[ x^4 - 8x^3 - x^2 + 68x - 84 = 0 \]

### Step 4: Use Trial and Error to Find Roots
We will now use trial and error to find rational roots. We will test the options provided: -1, -2, -3, 7, -3, and 2.

**Testing x = -1:**

\[ (-1 + 2)(-1 - 5)(-1 - 6)(-1 + 1) = 1 \cdot (-6) \cdot (-7) \cdot 0 = 0 \quad \text{(not equal to 144)} \]

**Testing x = -2:**

\[ (-2 + 2)(-2 - 5)(-2 - 6)(-2 + 1) = 0 \quad \text{(not equal to 144)} \]

**Testing x = -3:**

\[ (-3 + 2)(-3 - 5)(-3 - 6)(-3 + 1) = (-1)(-8)(-9)(-2) = -144 \quad \text{(not equal to 144)} \]

**Testing x = 7:**

\[ (7 + 2)(7 - 5)(7 - 6)(7 + 1) = 9 \cdot 2 \cdot 1 \cdot 8 = 144 \quad \text{(equal to 144)} \]

### Step 5: Confirm Other Options
**Testing x = -3 again:**

\[ (-3 + 2)(-3 - 5)(-3 - 6)(-3 + 1) = (-1)(-8)(-9)(-2) = -144 \quad \text{(not equal to 144)} \]

**Testing x = 2:**

\[ (2 + 2)(2 - 5)(2 - 6)(2 + 1) = 4 \cdot (-3) \cdot (-4) \cdot 3 = 144 \quad \text{(equal to 144)} \]

### Conclusion
The values of \(x\) that satisfy the equation are \(7\) and \(2\). Therefore, the correct option is:

**Answer: 7, -3, 2**