In each of the following cases (Q.9-12), find whether g(x) is a factor of p(x) :
`p(x)=x^(4)+3x^(2)-4, " " g(x)=x+2`

Answer

To determine whether \( g(x) = x + 2 \) is a factor of \( p(x) = x^4 + 3x^2 - 4 \), we can use the Factor Theorem. According to the Factor Theorem, \( g(x) \) is a factor of \( p(x) \) if and only if \( p(-2) = 0 \).

### Step-by-Step Solution:

1. **Identify the polynomial and the factor:**
- We have \( p(x) = x^4 + 3x^2 - 4 \) and \( g(x) = x + 2 \).

2. **Set \( g(x) = 0 \) to find the value of \( x \):**
- Solve \( x + 2 = 0 \).
- This gives us \( x = -2 \).

3. **Substitute \( x = -2 \) into \( p(x) \):**
- Calculate \( p(-2) \):
\[ p(-2) = (-2)^4 + 3(-2)^2 - 4 \]

4. **Calculate each term:**
- \( (-2)^4 = 16 \)
- \( (-2)^2 = 4 \), so \( 3(-2)^2 = 3 \times 4 = 12 \)
- Now substitute these values back into the equation:
\[ p(-2) = 16 + 12 - 4 \]

5. **Simplify the expression:**
- Combine the values:
\[ p(-2) = 16 + 12 = 28 \]
- Then subtract 4:
\[ p(-2) = 28 - 4 = 24 \]

6. **Determine if \( g(x) \) is a factor:**
- Since \( p(-2) = 24 \) and not equal to 0, \( g(x) = x + 2 \) is **not** a factor of \( p(x) \).

### Conclusion:
Thus, \( g(x) = x + 2 \) is not a factor of \( p(x) = x^4 + 3x^2 - 4 \).