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The given quadratic equation is \( ax^2 + bx + c = 0 \) with roots \( -4 \) and \( \frac{3}{2} \). We are asked to find the value of \( a - b + c \) given that \( a \), \( b \), and \( c \) are integers.

### Step 1: Use the relationship between the roots and the coefficients
For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the sum and product of the roots are related to the coefficients by Vieta's formulas:

- Sum of roots = \( -\frac{b}{a} \)
- Product of roots = \( \frac{c}{a} \)

### Step 2: Apply Vieta's formulas with the given roots
- The sum of the roots is \( -4 + \frac{3}{2} = -\frac{5}{2} \). Therefore,
  \[
  -\frac{b}{a} = -\frac{5}{2}
  \]
  which gives \( b = \frac{5a}{2} \).

- The product of the roots is \( -4 \times \frac{3}{2} = -6 \). Therefore,
  \[
  \frac{c}{a} = -6
  \]
  which gives \( c = -6a \).

### Step 3: Solve for \( a \), \( b \), and \( c \)
Since \( b = \frac{5a}{2} \) must be an integer, \( a \) must be even. Let \( a = 2 \), the smallest even integer. Then:

- \( b = \frac{5(2)}{2} = 5 \)
- \( c = -6(2) = -12 \)

### Step 4: Calculate \( a - b + c \)
Now we compute \( a - b + c \):
\[
a - b + c = 2 - 5 - 12 = -15
\]

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Given that the quadratic equation ax^2 + bx + c = 0 has roots -4 and 3/2, where a, b, and c are integers, find the value of a - b + c.

This quadratic equation problem involves finding the value of a - b + c, where a, b, and c are integers. The equation has roots -4 and 3/2, and key algebraic concepts like Vieta's formulas are used to solve the problem. Suitable for high school students, Grades 10-12.

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