To factorize 2x\(^2\) - 21x + 45, we need to find two numbers that multiply to give 2\(\times\)45 = 90 and add to give -21. Let's factorize 90 to find such numbers:
90 = 1\(\times\)90 = 2\(\times\)45 = 3\(\times\)30 = 5\(\times\)18 = 6\(\times\)15
Out of these, the pair that adds up to -21 is 6 and 15. So, we can rewrite the expression as:
2x\(^2\) - 21x + 45 = 2x\(^2\) - 6x - 15x + 45
Now, we can group the first two terms and the last two terms separately and factorize them using the distributive law. That gives us:
2x\(^2\) - 6x - 15x + 45 = 2x(x - 3) - 15(x - 3)
Notice that we have a common factor of (x - 3) in both terms. We can factor it out to get the final answer:
2x\(^2\) - 21x + 45 = (x - 3)(2x - 15)
Therefore, the correct answer is (x - 3)(2x - 15).