Find the HCF of the polynomials `f(x)= 6(x^(3) +3x^(2)) (x^(2)-16) (x^(2) + 9x + 18) and g(x) = 8(x^(4) + 4x^(3)) (x^(2) + 6x + 9)^(2)`

Answer

To find the HCF (Highest Common Factor) of the given polynomials \( f(x) \) and \( g(x) \), we will follow these steps:

### Step 1: Factor the first polynomial \( f(x) \)

Given:
\[ f(x) = 6(x^3 + 3x^2)(x^2 - 16)(x^2 + 9x + 18) \]

1. **Factor out the coefficients and common terms:**
- \( 6 = 2 \cdot 3 \)
- From \( x^3 + 3x^2 \), we can factor out \( x^2 \):
\[ x^3 + 3x^2 = x^2(x + 3) \]
- \( x^2 - 16 \) can be factored as a difference of squares:
\[ x^2 - 16 = (x - 4)(x + 4) \]
- For \( x^2 + 9x + 18 \), we can factor it as:
\[ x^2 + 9x + 18 = (x + 3)(x + 6) \]

2. **Combine all factors:**
\[ f(x) = 6 \cdot x^2 \cdot (x + 3) \cdot (x - 4) \cdot (x + 4) \cdot (x + 3)(x + 6) \]
\[ = 6x^2(x + 3)^2(x - 4)(x + 4)(x + 6) \]

### Step 2: Factor the second polynomial \( g(x) \)

Given:
\[ g(x) = 8(x^4 + 4x^3)(x^2 + 6x + 9)^2 \]

1. **Factor out the coefficients and common terms:**
- \( 8 = 2^3 \)
- From \( x^4 + 4x^3 \), we can factor out \( x^3 \):
\[ x^4 + 4x^3 = x^3(x + 4) \]
- For \( (x^2 + 6x + 9)^2 \), we can recognize it as a perfect square:
\[ x^2 + 6x + 9 = (x + 3)^2 \]
- Therefore:
\[ (x^2 + 6x + 9)^2 = ((x + 3)^2)^2 = (x + 3)^4 \]

2. **Combine all factors:**
\[ g(x) = 8 \cdot x^3 \cdot (x + 4) \cdot (x + 3)^4 \]
\[ = 2^3 \cdot x^3 \cdot (x + 4) \cdot (x + 3)^4 \]

### Step 3: Find the HCF of \( f(x) \) and \( g(x) \)

Now we have:
- \( f(x) = 6x^2(x + 3)^2(x - 4)(x + 4)(x + 6) \)
- \( g(x) = 2^3 x^3 (x + 4)(x + 3)^4 \)

1. **Identify the common factors:**
- Coefficient: \( 6 = 2 \cdot 3 \) and \( 8 = 2^3 \) → HCF of coefficients is \( 2^1 = 2 \)
- \( x^2 \) from \( f(x) \) and \( x^3 \) from \( g(x) \) → HCF is \( x^2 \)
- \( (x + 3)^2 \) from \( f(x) \) and \( (x + 3)^4 \) from \( g(x) \) → HCF is \( (x + 3)^2 \)
- \( (x + 4) \) is common in both.
- \( (x - 4) \) and \( (x + 6) \) are not common in \( g(x) \).

2. **Combine the common factors:**
\[ \text{HCF} = 2 \cdot x^2 \cdot (x + 3)^2 \cdot (x + 4) \]

### Final Answer:
\[ \text{HCF} = 2x^2(x + 3)^2(x + 4) \]

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