Step-by-step explanation:

1. Assume √2 and 3 - 4√5 are both rational numbers.

2. Since they are rational, we can express them as fractions: √2 = a/b and 3 - 4√5 = c/d, where a, b, c, and d are integers with no common factors.

3. Add the two fractions: (a/b) + (c/d) = (ad + bc) / (bd).

4. The sum is also a rational number, as the numerator and denominator are integers.

5. However, √2 + (3 - 4√5) = 3 + √2 - 4√5, which simplifies to √2 - 4√5.

6. This sum is irrational, as it cannot be expressed as a fraction.

7. This contradicts our initial assumption, proving that either √2 or 3 - 4√5 must be irrational.

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