Find the LCM of the polynomials
`f(x)= 4(x-1)^(2) (x+1)^(2) (x^(2) + 6x + 8) and g(x)= 10 (x-1) (x-2) (x^(2) + 7x + 10)`

Answer

To find the LCM of the polynomials \( f(x) \) and \( g(x) \), we will follow these steps:

### Step 1: Factor the polynomials
First, we need to express both polynomials in their factored form.

**For \( f(x) \):**
\[ f(x) = 4(x-1)^2 (x+1)^2 (x^2 + 6x + 8) \]
We can factor \( x^2 + 6x + 8 \):
\[ x^2 + 6x + 8 = (x+2)(x+4) \]
Thus, we can rewrite \( f(x) \):
\[ f(x) = 4(x-1)^2 (x+1)^2 (x+2)(x+4) \]

**For \( g(x) \):**
\[ g(x) = 10(x-1)(x-2)(x^2 + 7x + 10) \]
We can factor \( x^2 + 7x + 10 \):
\[ x^2 + 7x + 10 = (x+2)(x+5) \]
Thus, we can rewrite \( g(x) \):
\[ g(x) = 10(x-1)(x-2)(x+2)(x+5) \]

### Step 2: Identify the LCM of the coefficients
Now, we find the LCM of the coefficients of \( f(x) \) and \( g(x) \):
- Coefficient of \( f(x) \) is 4.
- Coefficient of \( g(x) \) is 10.

The LCM of 4 and 10 is:
\[ \text{LCM}(4, 10) = 20 \]

### Step 3: Identify the LCM of the polynomial factors
Next, we identify the highest powers of each factor from both polynomials:

- **For \( (x-1) \)**:
- \( f(x) \) has \( (x-1)^2 \)
- \( g(x) \) has \( (x-1)^1 \)
- Take \( (x-1)^2 \)

- **For \( (x-2) \)**:
- \( f(x) \) does not have \( (x-2) \)
- \( g(x) \) has \( (x-2)^1 \)
- Take \( (x-2)^1 \)

- **For \( (x+2) \)**:
- Both \( f(x) \) and \( g(x) \) have \( (x+2)^1 \)
- Take \( (x+2)^1 \)

- **For \( (x+4) \)**:
- \( f(x) \) has \( (x+4)^1 \)
- \( g(x) \) does not have \( (x+4) \)
- Take \( (x+4)^1 \)

- **For \( (x+5) \)**:
- \( f(x) \) does not have \( (x+5) \)
- \( g(x) \) has \( (x+5)^1 \)
- Take \( (x+5)^1 \)

### Step 4: Combine all parts to find the LCM
Now, we can combine the LCM of the coefficients and the LCM of the polynomial factors:
\[ \text{LCM}(f(x), g(x)) = 20 \cdot (x-1)^2 \cdot (x-2) \cdot (x+2) \cdot (x+4) \cdot (x+5) \]

Thus, the final answer is:
\[ \text{LCM}(f(x), g(x)) = 20(x-1)^2(x-2)(x+2)(x+4)(x+5) \]