Okay, let's see. I need to evaluate (1 + 2i) multiplied by 6 and then subtract 3i. Hmm, so first, I guess I should handle the multiplication part. Let me recall how to multiply complex numbers. Oh right, when you multiply two complex numbers, you distribute each term. So (a + bi)(c + di) = ac + adi + bci + bdi². But here, the second term is just 6, which is a real number. So maybe it's easier to just multiply each term in the complex number by 6. 

Wait, the problem is (1 + 2i) multiplied by 6 minus 3i. So first step is to multiply (1 + 2i) by 6. Let me do that. 6 times 1 is 6, and 6 times 2i is 12i. So that gives 6 + 12i. Then, subtract 3i from that result. So 6 + 12i - 3i. Combining like terms, 12i - 3i is 9i. So the final result should be 6 + 9i. 

Wait, that seems straightforward, but let me double-check. Let me verify each step again. First, multiplying (1 + 2i) by 6. Multiplying a complex number by a real number just scales both the real and imaginary parts. So 6*1 = 6, 6*2i = 12i. Then subtract 3i. 12i - 3i is indeed 9i. So total is 6 + 9i. 

Alternatively, maybe I can write it out step by step. Let's break it down:

Original expression: (1 + 2i) * 6 - 3i

First, compute (1 + 2i) * 6:

Multiply each term by 6:

1*6 = 6

2i*6 = 12i

So that gives 6 + 12i.

Then subtract 3i:

6 + 12i - 3i = 6 + (12i - 3i) = 6 + 9i.

Yep, that's the same result. 

Alternatively, if I use the distributive property for the entire expression, maybe I can see it another way. But since 6 is a real number, multiplying it by the complex number is straightforward. Subtracting 3i is just adding the negative 3i. 

Another way to check: Let's compute the entire expression as (1*6 + 2i*6) - 3i, which is the same as (6 + 12i) - 3i. Then combine the imaginary parts: 12i - 3i = 9i, so 6 + 9i. 

Alternatively, if I factor out the 3i somehow, but that might complicate things. Let me think. Wait, maybe not necessary. The straightforward method seems correct. 

Is there any chance I made a mistake in the signs? Let's see. The original problem is (1 + 2i)6 - 3i. So the first term is positive 6, second term is positive 12i, then subtract 3i. So 12i minus 3i is 9i. Correct. 

Alternatively, if I had a different sign, like if it was (1 + 2i)6 - (-3i), that would be different, but here it's just minus 3i. 

Alternatively, maybe I can write it as 6*(1 + 2i) - 3i. Then, 6*(1 + 2i) is 6 + 12i, then subtract 3i. Same result. 

Another check: Let's compute the real and imaginary parts separately. The real part is 6 (from 6*1) and then there's no other real part from the subtraction. The imaginary part is 12i (from 6*2i) minus 3i, which is 9i. So real part is 6, imaginary part is 9, so 6 + 9i. 

Alternatively, if I use the formula for complex multiplication: (a + bi)(c + di) = (ac - bd) + (ad + bc)i. Wait, but here the second factor is 6, which is (6 + 0i). So (1 + 2i)(6 + 0i) = (1*6 - 2i*0) + (1*0 + 2i*6)i. Wait, that seems a bit confusing. Let me apply the formula correctly. 

The formula is (a + bi)(c + di) = ac + adi + bci + bdi². Since i² = -1, so it becomes (ac - bd) + (ad + bc)i. 

But here, the second term is 6, which is 6 + 0i, so c = 6, d = 0. So applying the formula:

Real part: a*c - b*d = 1*6 - 2*0 = 6 - 0 = 6

Imaginary part: a*d + b*c = 1*0 + 2*6 = 0 + 12 = 12

So the product is 6 + 12i. Then subtract 3i: 6 + 12i - 3i = 6 + 9i. 

Yes, same result. 

Alternatively, I can use rectangular form addition. Let me try that. 

First, compute (1 + 2i)*6. Since 6 is a real number, scaling the complex number by 6 scales both the real and imaginary parts. So 1*6 = 6, 2i*6 = 12i. Then subtract 3i: 12i - 3i = 9i. So total is 6 + 9i. 

Alternatively, maybe write the entire expression as (1*6 + 2i*6) - 3i = (6 + 12i) - 3i = 6 + (12i - 3i) = 6 + 9i. 

All methods lead to the same result. 

Wait, maybe I can check using decimal equivalents? But since all numbers are integers, that might not be necessary. Let me see. 

Alternatively, if I convert to polar form, but that might be more complicated. Let's see. 

The complex number (1 + 2i) can be converted to polar form. The magnitude is sqrt(1² + 2²) = sqrt(5), and the angle is arctan(2/1) = arctan(2). Then multiplying by 6 would scale the magnitude by 6, so 6*sqrt(5), and the angle remains arctan(2). Then subtract 3i. Hmm, subtracting 3i is equivalent to adding -3i, which in polar form would be a complex number with magnitude 3 and angle -90 degrees (since it's along the negative imaginary axis). 

But adding two complex numbers in polar form is not straightforward. It's easier to convert them to rectangular form and then add/subtract. So maybe polar form isn't helpful here. 

Alternatively, using Euler's formula? Probably not necessary here. 

So, given all these checks, I think the answer is 6 + 9i. 

Just to recap:

Original expression: (1 + 2i)*6 - 3i

Multiply out: 6*1 + 6*2i - 3i = 6 + 12i - 3i = 6 + 9i.

Yes, that's correct. 

I don't see any mistakes in the steps. All methods lead to the same result. So the final answer should be 6 + 9i, which in boxed form is \boxed{6 + 9i}.
Therefore, the evaluated result of $(1+2i)6-3i$ is $\boxed{6 + 9i}$.
</think>