Find all values of 'm' which `(2m-3)x^(2)+2mx+4 lt 0` for all real x.

Answer

To find all values of \( m \) such that the quadratic expression \( (2m-3)x^2 + 2mx + 4 < 0 \) for all real \( x \), we need to analyze the conditions under which a quadratic function is negative for all \( x \).

### Step-by-step Solution:

1. **Identify the coefficients**:
The quadratic expression can be written in the standard form \( ax^2 + bx + c \), where:
- \( a = 2m - 3 \)
- \( b = 2m \)
- \( c = 4 \)

2. **Condition for the quadratic to be negative**:
For the quadratic \( ax^2 + bx + c < 0 \) to hold for all \( x \):
- The coefficient \( a \) must be less than 0 (i.e., the parabola opens downwards).
- The discriminant \( D \) must be less than 0 (i.e., there are no real roots).

3. **Set the conditions**:
- From \( a < 0 \):
\[ 2m - 3 < 0 \implies 2m < 3 \implies m < \frac{3}{2} \]
- From the discriminant \( D < 0 \):
The discriminant \( D \) is given by \( D = b^2 - 4ac \):
\[ D = (2m)^2 - 4(2m - 3)(4) \]
Simplifying this:
\[ D = 4m^2 - 16(2m - 3) = 4m^2 - 32m + 48 \]
Factor out 4:
\[ D = 4(m^2 - 8m + 12) \]
Now, we need \( m^2 - 8m + 12 < 0 \).

4. **Solve the quadratic inequality**:
To find the roots of \( m^2 - 8m + 12 = 0 \):
\[ m = \frac{8 \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot 12}}{2 \cdot 1} = \frac{8 \pm \sqrt{64 - 48}}{2} = \frac{8 \pm \sqrt{16}}{2} = \frac{8 \pm 4}{2} \]
This gives us:
\[ m = \frac{12}{2} = 6 \quad \text{and} \quad m = \frac{4}{2} = 2 \]
The roots are \( m = 2 \) and \( m = 6 \).

5. **Determine the intervals**:
The quadratic \( m^2 - 8m + 12 \) opens upwards (as the coefficient of \( m^2 \) is positive). The expression is negative between its roots:
\[ 2 < m < 6 \]

6. **Combine the conditions**:
We have two conditions:
- From \( a < 0 \): \( m < \frac{3}{2} \)
- From \( D < 0 \): \( 2 < m < 6 \)

The first condition \( m < \frac{3}{2} \) does not overlap with the second condition \( 2 < m < 6 \). Therefore, there are no values of \( m \) that satisfy both conditions simultaneously.

### Conclusion:
There are no values of \( m \) such that \( (2m-3)x^2 + 2mx + 4 < 0 \) for all real \( x \).