Okay, so I need to combine these two fractions into a single fraction. Let me think about how to do that. The expression is (4 + 6a)/5 minus (1 + 3a)/4. Hmm, right, to subtract these two fractions, I need a common denominator. The denominators are 5 and 4, so the least common denominator (LCD) would be their least common multiple. Let me recall, the LCM of 5 and 4... 5 is prime, 4 is 2 squared. So the LCM is 5 times 4, which is 20. So the LCD is 20. 

Alright, so I need to convert both fractions to have 20 as the denominator. Let's start with the first fraction: (4 + 6a)/5. To convert the denominator from 5 to 20, I need to multiply both numerator and denominator by 4. So that would give me (4 + 6a)*4 / (5*4) = (16 + 24a)/20. Let me check that: 4*4 is 16, 6a*4 is 24a, and 5*4 is 20. Yep, that's correct.

Now the second fraction: (1 + 3a)/4. To convert denominator from 4 to 20, multiply numerator and denominator by 5. So that becomes (1 + 3a)*5 / (4*5) = (5 + 15a)/20. Checking: 1*5 is 5, 3a*5 is 15a, 4*5 is 20. Right.

Now the original expression becomes (16 + 24a)/20 - (5 + 15a)/20. Since the denominators are the same, I can combine the numerators: [16 + 24a - 5 - 15a]/20. Let me compute the numerator step by step. 

First, combine the constant terms: 16 - 5 = 11. Then the terms with 'a': 24a - 15a = 9a. So the numerator becomes 11 + 9a. Therefore, the combined fraction is (11 + 9a)/20. Wait, but usually we write the terms with the variable first. So maybe it's better to write it as (9a + 11)/20. Either way is correct, but often the variable term comes first. Let me confirm the arithmetic again.

Original numerators after conversion: 16 +24a and 5 +15a. Subtracting the second from the first: 16 -5 is 11, 24a -15a is 9a. So yes, 11 +9a. So the final expression is (9a +11)/20. Let me check once more to be sure.

Alternatively, maybe I made a mistake in combining the numerators. Let's go through it again. First fraction after conversion: 16 +24a. Second fraction: 5 +15a. Subtracting the second from the first: (16 -5) + (24a -15a) = 11 +9a. Yes, correct. So the numerator is 11 +9a. Therefore, the single fraction is (11 +9a)/20. 

Alternatively, factoring, maybe? Let's see. 11 and 9 have a common factor? 11 is prime, 9 is 3 squared. No common factors, so we can't simplify further. So that's the simplest form. 

Wait, but let me check if the initial step of finding the LCD was correct. The denominators are 5 and 4. The LCM of 5 and 4 is indeed 20. So that step is correct. Then converting both fractions properly. Let me verify the multiplication:

First fraction: (4 +6a)/5. Multiply numerator and denominator by 4: 4*4=16, 6a*4=24a, denominator 5*4=20. Correct.

Second fraction: (1 +3a)/4. Multiply numerator and denominator by 5: 1*5=5, 3a*5=15a, denominator 4*5=20. Correct.

Subtraction: (16 +24a) - (5 +15a) = 16 -5 +24a -15a = 11 +9a. Correct. So numerator is 11 +9a. Denominator 20. So final answer is (9a +11)/20. 

Alternatively, writing it as (11 +9a)/20 is also correct. Depending on the preference, but usually the numerical term comes first. So the answer is (9a +11)/20. Let me check once more with another approach. Let's expand the original expression and see if the steps are correct.

Original expression: (4 +6a)/5 - (1 +3a)/4. Let's compute each term separately:

First term: (4 +6a)/5 = 4/5 + (6a)/5. Second term: (1 +3a)/4 = 1/4 + (3a)/4. Then subtract the second term from the first: (4/5 +6a/5) - (1/4 +3a/4). Distribute the subtraction: 4/5 -1/4 +6a/5 -3a/4. Now combine the constants and the coefficients of 'a' separately.

Constants: 4/5 -1/4. Let's compute that. LCD is 20. 4/5 is 16/20, 1/4 is 5/20. So 16/20 -5/20 = 11/20.

Coefficients of 'a': 6a/5 -3a/4. Let's compute that. LCD is 20. 6a/5 is (6a*4)/20 =24a/20. 3a/4 is (3a*5)/20=15a/20. So 24a/20 -15a/20 =9a/20.

Therefore, combining constants and coefficients: 11/20 +9a/20 = (9a +11)/20. Same result. So that confirms it. Therefore, the answer is (9a +11)/20. 

I think that's solid. Both methods lead to the same result, so I can be confident that it's correct. No mistakes detected in the process.
Therefore, the expression simplifies to $\boxed{\frac{9a + 11}{20}}$.
</think>