For all `' x^(prime),x^2+2a x+(10-3a)>0,` then the interval in which `' a '` lies is (2004, 1M) `a<-5` (b) `-55` (d) `2

Options

A. ` a le -5`
B. `-5 lt a lt 2 `
C. `a gt 5`
D. `2 lt a lt 5`

Answer

To solve the inequality \( x^2 + 2ax + (10 - 3a) > 0 \) for all \( x \), we need to ensure that the quadratic expression is always positive. This can be achieved if the coefficient of \( x^2 \) is positive and the discriminant of the quadratic is negative.

### Step 1: Identify the coefficients
The given quadratic is:
\[ x^2 + 2ax + (10 - 3a) \]
Here, the coefficients are:
- \( A = 1 \) (coefficient of \( x^2 \))
- \( B = 2a \) (coefficient of \( x \))
- \( C = 10 - 3a \)

### Step 2: Check the condition for \( A \)
Since \( A = 1 \) is positive, this condition is satisfied.

### Step 3: Find the discriminant
The discriminant \( D \) of a quadratic equation \( Ax^2 + Bx + C \) is given by:
\[ D = B^2 - 4AC \]
Substituting the values:
\[ D = (2a)^2 - 4(1)(10 - 3a) \]
This simplifies to:
\[ D = 4a^2 - 40 + 12a \]
Rearranging gives:
\[ D = 4a^2 + 12a - 40 \]

### Step 4: Set the discriminant less than zero
For the quadratic to be always positive, we need:
\[ 4a^2 + 12a - 40 < 0 \]
Dividing the entire inequality by 4:
\[ a^2 + 3a - 10 < 0 \]

### Step 5: Factor the quadratic
To factor \( a^2 + 3a - 10 \), we look for two numbers that multiply to \(-10\) and add to \(3\). The factors are \(5\) and \(-2\):
\[ (a + 5)(a - 2) < 0 \]

### Step 6: Determine the intervals
Now we need to find the intervals where the product is negative. The critical points are \( a = -5 \) and \( a = 2 \).

Using a sign chart or the wavy curve method:
- For \( a < -5 \): Both factors are negative, so the product is positive.
- For \( -5 < a < 2 \): One factor is positive and the other is negative, so the product is negative.
- For \( a > 2 \): Both factors are positive, so the product is positive.

Thus, the inequality \( (a + 5)(a - 2) < 0 \) holds true for:
\[ -5 < a < 2 \]

### Final Answer
The interval in which \( a \) lies is:
\[ \boxed{-5 < a < 2} \]