The sum of all intergral values of `lamda` for which `(lamda^(2) + lamda -2) x ^(2) + (lamda +2) x lt 1 AA x in R,` is:

Options

A. `-1`
B. `-3`
C. 0
D. `-2`

Answer

To solve the problem, we need to find the sum of all integral values of \( \lambda \) for which the inequality

\[ (\lambda^2 + \lambda - 2)x^2 + (\lambda + 2)x < 1 \]

holds for all real numbers \( x \).

### Step 1: Rearranging the Inequality
We start by rearranging the inequality:

\[ (\lambda^2 + \lambda - 2)x^2 + (\lambda + 2)x - 1 < 0 \]

This is a quadratic inequality in \( x \).

### Step 2: Identifying Coefficients
Let \( a = \lambda^2 + \lambda - 2 \), \( b = \lambda + 2 \), and \( c = -1 \). The quadratic can be expressed as:

\[ ax^2 + bx + c < 0 \]

### Step 3: Conditions for the Quadratic to be Negative
For the quadratic \( ax^2 + bx + c < 0 \) to hold for all \( x \), the following conditions must be satisfied:
1. \( a < 0 \) (the parabola opens downwards).
2. The discriminant \( D = b^2 - 4ac < 0 \) (there are no real roots).

### Step 4: Finding Conditions on \( a \)
We need to solve \( a < 0 \):

\[ \lambda^2 + \lambda - 2 < 0 \]

Factoring the quadratic:

\[ (\lambda - 1)(\lambda + 2) < 0 \]

### Step 5: Solving the Inequality
The critical points are \( \lambda = -2 \) and \( \lambda = 1 \). We test intervals determined by these points:

- For \( \lambda < -2 \): Choose \( \lambda = -3 \) → \( (-3 - 1)(-3 + 2) = (-4)(-1) > 0 \)
- For \( -2 < \lambda < 1 \): Choose \( \lambda = 0 \) → \( (0 - 1)(0 + 2) = (-1)(2) < 0 \)
- For \( \lambda > 1 \): Choose \( \lambda = 2 \) → \( (2 - 1)(2 + 2) = (1)(4) > 0 \)

Thus, the solution to \( (\lambda - 1)(\lambda + 2) < 0 \) is:

\[ -2 < \lambda < 1 \]

### Step 6: Finding Integral Values of \( \lambda \)
The integral values of \( \lambda \) in the interval \( (-2, 1) \) are \( -1 \) and \( 0 \).

### Step 7: Summing Integral Values
Now, we sum the integral values:

\[ -1 + 0 = -1 \]

### Final Answer
The sum of all integral values of \( \lambda \) for which the inequality holds is:

\[ \boxed{-1} \]