Factors of 11a^2 + 54a + 63:

To find the factors of the given expression, we need to factorize it completely.

1. Common factor:
The first step is to check if there is a common factor among the three terms. In this case, we can see that 9 is a common factor of all the terms. So, we can factor out 9 from the expression.

9(11a^2 + 6a + 7)

2. Factoring the quadratic expression:
Next, we need to factorize the quadratic expression inside the parentheses. To do this, we look for two numbers that multiply to give the coefficient of the quadratic term (11) and add up to give the coefficient of the linear term (6). In this case, the numbers are 3 and 2.

9(11a^2 + 3a + 2a + 7)

3. Grouping:
Now, we group the terms together and factor by grouping.

9((11a^2 + 3a) + (2a + 7))

4. Factoring out common factors from each group:
In each group, we can factor out the common factors.

9(a(11a + 3) + 1(2a + 7))

5. Factoring the remaining expressions:
Now, we can factor the expressions inside the parentheses.

9(a(11a + 3) + 1(2a + 7))
9(a(11a + 3) + (2a + 7))

The factors of the given expression are (11a + 3) and (2a + 7).

6. Combining the factors:
Finally, we can write the factors of the expression by combining the factors we obtained.

(11a + 3)(2a + 7)

Therefore, the correct answer is option 'A': (11a + 3)(2a + 7).