The HCF of two expressions is x and their LCM is `x^3 – 9x`.If one of the expressions is `x^2+ 3x`, then find the other expression.

Answer

To find the other expression given the HCF and LCM of two expressions, we can use the relationship between them. The product of the HCF and LCM of two expressions is equal to the product of the two expressions.

### Step-by-step Solution:

1. **Identify the Given Values**:
- HCF (Highest Common Factor) = \( x \)
- LCM (Lowest Common Multiple) = \( x^3 - 9x \)
- One of the expressions (let's call it \( p_1 \)) = \( x^2 + 3x \)
- The other expression (let's call it \( p_2 \)) is what we need to find.

2. **Use the Relationship**:
The relationship we use is:
\[ \text{HCF} \times \text{LCM} = p_1 \times p_2 \]
Substituting the known values:
\[ x \times (x^3 - 9x) = (x^2 + 3x) \times p_2 \]

3. **Simplify the Left Side**:
First, simplify the left side:
\[ x \times (x^3 - 9x) = x^4 - 9x^2 \]

4. **Set Up the Equation**:
Now we have:
\[ x^4 - 9x^2 = (x^2 + 3x) \times p_2 \]

5. **Factor Out \( p_2 \)**:
To find \( p_2 \), we can rearrange the equation:
\[ p_2 = \frac{x^4 - 9x^2}{x^2 + 3x} \]

6. **Factor the Numerator**:
Factor \( x^4 - 9x^2 \):
\[ x^4 - 9x^2 = x^2(x^2 - 9) = x^2(x - 3)(x + 3) \]

7. **Factor the Denominator**:
Factor \( x^2 + 3x \):
\[ x^2 + 3x = x(x + 3) \]

8. **Substitute Back into \( p_2 \)**:
Now substituting the factors back into the equation for \( p_2 \):
\[ p_2 = \frac{x^2(x - 3)(x + 3)}{x(x + 3)} \]

9. **Cancel Common Terms**:
Cancel \( x + 3 \) from the numerator and denominator:
\[ p_2 = \frac{x^2(x - 3)}{x} = x(x - 3) \]

10. **Final Expression for \( p_2 \)**:
Thus, the other expression \( p_2 \) is:
\[ p_2 = x^2 - 3x \]

### Final Answer:
The other expression is \( x^2 - 3x \).