Mind you I'm not an expert on mathematical proofs but here's a simple take on it:
Suppose that 3*(√2)-5 is a rational number, it can therefore be expressed as a ratio of integers m and n:
Adding 5 to both sides:
Dividing both sides by 3:
Now since both m and n are integers, then both terms (m+5n) and 3n are also integers which means √2 can then be expressed as a ratio of integers (ie it's a rational number). Since we are explicitly told that √2 is not a rational number and therefore cannot be expressed as a ratio of integers, this is obviously false. So therefore, the initial assumption that 3(√2)-5 is a rational number is also false so it must be an irrational number.