Find the interval in which .'m' lies so that the expression `(mx^(2) + 3x-4)/(-4x^(2)+3 x + m)` can take all real values, `x in R`.

Answer

To solve the problem of finding the interval in which \( m \) lies so that the expression

\[ \frac{mx^2 + 3x - 4}{-4x^2 + 3x + m} \]

can take all real values, we will follow these steps:

### Step 1: Set up the equation
Let

\[ y = \frac{mx^2 + 3x - 4}{-4x^2 + 3x + m} \]

Cross-multiplying gives us:

\[ y(-4x^2 + 3x + m) = mx^2 + 3x - 4 \]

Rearranging this, we get:

\[ (m + 4y)x^2 + (3 - 3y)x + (my + 4) = 0 \]

### Step 2: Analyze the quadratic equation
For the quadratic equation in \( x \) to have real roots for all \( y \), the discriminant must be greater than or equal to zero. The discriminant \( D \) of the quadratic equation \( ax^2 + bx + c = 0 \) is given by:

\[ D = b^2 - 4ac \]

Here, \( a = m + 4y \), \( b = 3 - 3y \), and \( c = my + 4 \).

### Step 3: Calculate the discriminant
The discriminant \( D \) is:

\[ D = (3 - 3y)^2 - 4(m + 4y)(my + 4) \]

Expanding this gives:

\[ D = (3 - 3y)^2 - 4(m + 4y)(my + 4) \]

### Step 4: Set the discriminant condition
For the expression to take all real values, we require:

\[ D \geq 0 \]

This leads to a quadratic inequality in \( y \).

### Step 5: Solve the quadratic inequality
We simplify the discriminant condition and find the conditions on \( m \):

1. The coefficient of \( y^2 \) must be positive:

\[ 9 + 16m > 0 \implies m > -\frac{9}{16} \]

2. The discriminant of the quadratic in \( y \) must be less than zero:

\[ (46 + 4m^2)^2 - 4(9 + 16m)(9 + 16m) < 0 \]

This leads to solving the inequality which simplifies to:

\[ (m + 4)^2(m - 7)(m - 1) < 0 \]

### Step 6: Analyze the intervals
Using the wavey curve method, we find the critical points \( -4, 1, 7 \) and analyze the sign of the expression in the intervals:

- For \( m < -4 \): Positive
- For \( -4 < m < 1 \): Negative
- For \( 1 < m < 7 \): Negative
- For \( m > 7 \): Positive

### Step 7: Combine results
From the conditions derived, we have:

1. \( m > -\frac{9}{16} \)
2. \( m \) must be in the interval \( (1, 7) \)

The intersection of these conditions gives us the final interval:

\[ m \in (1, 7) \]

### Final Answer:
The interval in which \( m \) lies so that the expression can take all real values is:

\[ \boxed{(1, 7)} \]