The number of roots of the equation `((x+2)(x-5))/((x-3)(x+6))=(x-2)/(x+4)` is

Answer

To find the number of roots of the equation

\[ \frac{(x+2)(x-5)}{(x-3)(x+6)} = \frac{x-2}{x+4} \]

we will follow these steps:

### Step 1: Cross-Multiply
We start by cross-multiplying to eliminate the fractions:

\[ (x+2)(x-5)(x+4) = (x-2)(x-3)(x+6) \]

### Step 2: Expand Both Sides
Now we will expand both sides of the equation.

**Left Side:**
\[ (x+2)(x-5)(x+4) = (x^2 - 3x - 10)(x + 4) \]
Expanding this gives:
\[ x^3 + 4x^2 - 3x^2 - 12x - 10x - 40 = x^3 + x^2 - 22x - 40 \]

**Right Side:**
\[ (x-2)(x-3)(x+6) = (x^2 - 5x + 6)(x - 2) \]
Expanding this gives:
\[ x^3 - 2x^2 - 5x^2 + 10x + 6x - 12 = x^3 - 7x^2 + 16x - 12 \]

### Step 3: Set the Equation to Zero
Now we set both sides equal to each other:

\[ x^3 + x^2 - 22x - 40 = x^3 - 7x^2 + 16x - 12 \]

Subtract \(x^3\) from both sides:

\[ x^2 - 22x - 40 = -7x^2 + 16x - 12 \]

Combine like terms:

\[ x^2 + 7x^2 - 22x - 16x - 40 + 12 = 0 \]

This simplifies to:

\[ 8x^2 - 38x - 28 = 0 \]

### Step 4: Simplify the Quadratic Equation
We can simplify this equation by dividing everything by 2:

\[ 4x^2 - 19x - 14 = 0 \]

### Step 5: Calculate the Discriminant
Next, we calculate the discriminant \(D\) of the quadratic equation:

\[ D = b^2 - 4ac = (-19)^2 - 4 \cdot 4 \cdot (-14) \]

Calculating this gives:

\[ D = 361 + 224 = 585 \]

### Step 6: Determine the Number of Roots
Since the discriminant \(D > 0\), this means that the quadratic equation has two distinct real roots.

### Conclusion
Thus, the number of roots of the given equation is **2**.

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