`a, b, c, in R, a ne 0` and the quadratic equation `ax^(2) + bx + c = 0` has no real roots, then which one of the following is not true?

Options

A. `a + b + c gt 0`
B. `a(a+b+c)gt 0`
C. `ac(a+b+c)gt 0`
D. `c(a+b+c)gt 0`

Explanation

To solve the problem, we need to analyze the conditions under which the quadratic equation \( ax^2 + bx + c = 0 \) has no real roots. This occurs when the discriminant \( D \) is less than zero. The discriminant is given by:

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

Since we are given that \( D < 0 \), we can conclude that:

\[ b^2 - 4ac < 0 \implies b^2 < 4ac \]

This means that \( ac \) must be positive, as \( b^2 \) is always non-negative (since it is a square).

Next, we will analyze the options provided in the question to determine which one is not true.

### Step 1: Analyze the options

1. **Option A: \( a + b + c > 0 \)**

This option does not necessarily hold true. For example, if \( a = 1 \), \( b = -3 \), and \( c = 2 \), we have:
\[ a + b + c = 1 - 3 + 2 = 0 \]
This shows that \( a + b + c \) can be zero, which means it is not always greater than zero.

2. **Option B: \( a(a + b + c) > 0 \)**

Since \( a \) is not equal to zero, if \( a + b + c > 0 \), then this option holds true. However, if \( a + b + c \leq 0 \), this option may not hold.

3. **Option C: \( ac(a + b + c) > 0 \)**

Since we established that \( ac > 0 \), if \( a + b + c > 0 \), then this option is true. However, if \( a + b + c \leq 0 \), this option may not hold.

4. **Option D: \( c(a + b + c) > 0 \)**

This option also depends on the sign of \( c \) and \( a + b + c \). If \( c > 0 \) and \( a + b + c > 0 \), then this option holds true.

### Conclusion

From the analysis, we find that:

- **Option A** is not necessarily true since \( a + b + c \) can be zero or negative.
- **Options B, C, and D** can hold true under certain conditions.

Thus, the answer to the question is:

**Option A is not true.**