If the polynomial 6x^4 + 8x^3 - 5x^2 + ax + b is exactly divisible by 2x^2 - 5, then a = -20 and b = -25 to ensure a zero remainder.
If the polynomial 6x^4 + 8x^3 - 5x^2 + ax + b is exactly divisible by 2x^2 - 5, then the remainder when dividing these polynomials is zero. We can set up the division and equate the coefficients to find the values of a and b.
The long division looks like this:
```
3x^2 + 2x + 4
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2x^2 - 5 | 6x^4 + 8x^3 - 5x^2 + ax + b
- (6x^4 - 15x^2)
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8x^3 + 10x^2 + ax + b
- (8x^3 - 20x)
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10x^2 + (a + 20)x + b
- (10x^2 - 25)
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(a + 20)x + b + 25
```
For the remainder to be zero, the coefficients of x and the constant term must be zero:
1. Coefficient of x: a + 20 = 0 implies a = -20
2. Constant term: b + 25 = 0 implies b = -25
Therefore, the values of (a) and (b) are a = -20 and (b = -25).