Find the least value of `n` such that `(n 2)x^2+8x+n+4>0`,`AAx in R` ,where `n in N`.

Answer

To find the least value of \( n \) such that the inequality

\[ (n - 2)x^2 + 8x + (n + 4) > 0 \]

holds for all \( x \in \mathbb{R} \) and \( n \in \mathbb{N} \), we need to analyze the quadratic expression.

### Step 1: Identify the conditions for the quadratic to be positive for all \( x \)

For a quadratic expression \( ax^2 + bx + c \) to be positive for all \( x \):
1. The leading coefficient \( a \) must be greater than 0.
2. The discriminant \( D \) must be less than 0.

In our case:
- \( a = n - 2 \)
- \( b = 8 \)
- \( c = n + 4 \)

### Step 2: Set the condition for \( a > 0 \)

We need:

\[ n - 2 > 0 \implies n > 2 \]

### Step 3: Set the condition for the discriminant \( D < 0 \)

The discriminant \( D \) is given by:

\[ D = b^2 - 4ac = 8^2 - 4(n - 2)(n + 4) \]

Calculating \( D \):

\[ D = 64 - 4[(n - 2)(n + 4)] \]

Expanding the product:

\[ (n - 2)(n + 4) = n^2 + 4n - 2n - 8 = n^2 + 2n - 8 \]

Thus,

\[ D = 64 - 4(n^2 + 2n - 8) = 64 - 4n^2 - 8n + 32 \]

Simplifying gives:

\[ D = 96 - 4n^2 - 8n \]

Setting the discriminant less than zero:

\[ 96 - 4n^2 - 8n < 0 \]

### Step 4: Rearranging the inequality

Rearranging gives:

\[ 4n^2 + 8n - 96 > 0 \]

Dividing through by 4:

\[ n^2 + 2n - 24 > 0 \]

### Step 5: Factor the quadratic

Factoring the quadratic:

\[ (n + 6)(n - 4) > 0 \]

### Step 6: Determine the intervals

The critical points are \( n = -6 \) and \( n = 4 \). We analyze the sign of the expression in the intervals:

1. \( n < -6 \): Positive
2. \( -6 < n < 4 \): Negative
3. \( n > 4 \): Positive

Thus, the solution to the inequality \( (n + 6)(n - 4) > 0 \) is:

\[ n < -6 \quad \text{or} \quad n > 4 \]

### Step 7: Combine with the condition \( n > 2 \)

Since \( n \) must also be a natural number, we take the interval \( n > 4 \).

### Step 8: Find the least natural number satisfying the condition

The least natural number greater than 4 is:

\[ n = 5 \]

### Conclusion

Thus, the least value of \( n \) such that the inequality holds for all \( x \in \mathbb{R} \) is:

\[ \boxed{5} \]