When factoring a polynomial in the form ax^2 + bx-c, where a, b, and c are positive real numbers, should the signs in the binomials be both positive, negative, or one of each? Create an example to verify your claim.

1. Consider the polynomial \( ax^2 + bx - c \).
2. We need to factor it into binomials: \( (px + q)(rx + s) \).
3. Expanding the binomials:

(px + q)(rx + s) = prx^2 + (ps + qr)x + qs

4. Match the coefficients with the original polynomial:

a = pr, \quad b = ps + qr, \quad c = -qs

5. Since \( a \), \( b \), and \( c \) are positive, we need \( a \) and \( b \) terms to be positive while \( c \) is negative.
6. To satisfy these conditions, one binomial should have a positive constant, and the other should have a negative constant. Therefore, the signs in the binomials must be one positive and one negative.

Example:

1. Let's take \( 2x^2 + 3x - 4 \):
2. Factor it as \( (2x - 1)(x + 4) \):
3. Expand to verify:

(2x - 1)(x + 4) = 2x^2 + 8x - x - 4 = 2x^2 + 7x - 4

4. Verification shows it doesn't match \( 2x^2 + 3x - 4 \), let’s try another factor pair.
5. Factor it as \( (2x + 4)(x - 1) \):

(2x + 4)(x - 1) = 2x^2 - 2x + 4x - 4 = 2x^2 + 2x - 4

Again, try with different factor pairs \( (2x + 4)(x - 1) \)

[Step-by-Step Solution] One binomial's SIGN is positive and another is negative, final binomials for actual verification.