If `f(x)` is a cubic polynomial `x^3 + ax^2+ bx + c` such that `f(x)=0` has three distinct integral roots and `f(g(x)) = 0` does not have real roots, where `g(x) = x^2 + 2x - 5,` the minimum value of `a + b + c` is

Options

A. 504
B. 532
C. 719
D. 764

Explanation

To solve the problem, we need to find the minimum value of \( a + b + c \) for the cubic polynomial \( f(x) = x^3 + ax^2 + bx + c \) given that it has three distinct integral roots and that \( f(g(x)) = 0 \) does not have real roots, where \( g(x) = x^2 + 2x - 5 \).

### Step 1: Understand the roots of \( g(x) \)
First, we need to find the vertex of the quadratic function \( g(x) \):
\[ g(x) = x^2 + 2x - 5 \]
The vertex \( x \) value is given by:
\[ x = -\frac{b}{2a} = -\frac{2}{2} = -1 \]
Now, substituting \( x = -1 \) into \( g(x) \):
\[ g(-1) = (-1)^2 + 2(-1) - 5 = 1 - 2 - 5 = -6 \]
Thus, the vertex of \( g(x) \) is at \( (-1, -6) \) and it opens upwards.

### Step 2: Determine the condition for \( f(g(x)) = 0 \)
Since \( f(g(x)) = 0 \) does not have real roots, this means that the values of \( g(x) \) must be less than the minimum value of \( f(x) \). The roots of \( f(x) \) are denoted as \( \alpha, \beta, \gamma \) (the three distinct integral roots).

Given that \( g(x) \) reaches a minimum of \( -6 \) at \( x = -1 \), we require:
\[ \alpha, \beta, \gamma < -6 \]

### Step 3: Choose distinct integral roots
To minimize \( a + b + c \), we can choose the three distinct integral roots just below -6. The largest integers less than -6 are -7, -8, and -9.

### Step 4: Calculate \( a + b + c \)
Using Vieta's formulas, we know:
- The sum of the roots \( \alpha + \beta + \gamma = -a \)
- The sum of the products of the roots taken two at a time \( \alpha\beta + \beta\gamma + \gamma\alpha = b \)
- The product of the roots \( \alpha\beta\gamma = -c \)

Substituting \( \alpha = -7, \beta = -8, \gamma = -9 \):
1. Calculate \( a \):
\[ \alpha + \beta + \gamma = -7 - 8 - 9 = -24 \implies a = 24 \]
2. Calculate \( b \):
\[ \alpha\beta + \beta\gamma + \gamma\alpha = (-7)(-8) + (-8)(-9) + (-9)(-7) = 56 + 72 + 63 = 191 \implies b = 191 \]
3. Calculate \( c \):
\[ \alpha\beta\gamma = (-7)(-8)(-9) = -504 \implies c = 504 \]

### Step 5: Find \( a + b + c \)
Now, we can find:
\[ a + b + c = 24 + 191 + 504 = 719 \]

### Final Answer
Thus, the minimum value of \( a + b + c \) is:
\[ \boxed{719} \]