Okay, so I need to simplify the expression (1 + √2)/(2 + √3) and express it in the form A(1 + √B) - (√C + √D), where A, B, C, D are positive integers. Then find the sum A + B + C + D. Hmm, let me start by recalling how to simplify expressions with radicals in the denominator. Usually, the method is to rationalize the denominator by multiplying numerator and denominator by the conjugate of the denominator. 

The denominator here is 2 + √3. Its conjugate should be 2 - √3. So I'll multiply both the numerator and denominator by 2 - √3. Let me write that down:

[(1 + √2)(2 - √3)] / [(2 + √3)(2 - √3)]

First, let me compute the denominator. Using the difference of squares formula: (a + b)(a - b) = a² - b². So here, a is 2 and b is √3. Therefore, denominator becomes 2² - (√3)² = 4 - 3 = 1. Oh, that's nice! The denominator simplifies to 1, so the entire expression becomes just the numerator after simplification. Wait, no, actually, let me check that again. Wait, denominator is (2 + √3)(2 - √3) = 4 - (√3)² = 4 - 3 = 1. So denominator is 1. So then the expression is equal to the numerator: (1 + √2)(2 - √3). So I just need to multiply out the numerator.

Let me expand the numerator:

(1)(2) + (1)(-√3) + (√2)(2) + (√2)(-√3)

Calculating each term:

1*2 = 2

1*(-√3) = -√3

√2*2 = 2√2

√2*(-√3) = -√6

So combining all these terms:

2 - √3 + 2√2 - √6

So the entire expression simplifies to 2 - √3 + 2√2 - √6. Now, the problem says to express this in the form A(1 + √B) - (√C + √D). Let me see how I can rearrange the terms to match this form.

Looking at the simplified expression: 2 - √3 + 2√2 - √6. Let me group the terms with coefficients and the radicals:

2 + 2√2 - √3 - √6

Hmm, the desired form is A(1 + √B) - (√C + √D). Let's see. Let me factor out A from some terms. Since A is a positive integer, maybe A is 2? Let's try:

2(1 + √2) - (√3 + √6)

Yes, that works. Let me check:

2*(1) = 2

2*(√2) = 2√2

Then subtract √3 and √6:

2 + 2√2 - √3 - √6

Which is exactly what we have. So in this form, A is 2, B is 2, C is 3, D is 6. Therefore, A + B + C + D = 2 + 2 + 3 + 6 = 13.

Wait, but let me double-check to make sure I didn't make any mistakes. Let's go through the steps again.

Original expression: (1 + √2)/(2 + √3). Multiply numerator and denominator by (2 - √3):

Numerator: (1 + √2)(2 - √3) = 1*2 + 1*(-√3) + √2*2 + √2*(-√3) = 2 - √3 + 2√2 - √6. Denominator: (2 + √3)(2 - √3) = 4 - 3 = 1. So expression is 2 - √3 + 2√2 - √6. Then, factoring out 2 from the first two terms: 2(1 + √2) - (√3 + √6). So A=2, B=2, C=3, D=6. Sum is 2+2+3+6=13. That seems right.

Alternatively, maybe there's another way to factor the terms, but I think this is the simplest approach. Let me check if there's a different grouping. Suppose I tried to factor something else, but I don't think so. For example, if I tried to factor out a different A, but A has to be an integer. If A is 1, then we would have 2 - √3 + 2√2 - √6, which doesn't factor neatly. If A is 2, then the rest of the terms can be grouped into two separate radicals. So yes, that seems to be the correct approach.

Another way to verify is to plug the original expression and the simplified form into a calculator and check if they are equal. Let me compute the original expression numerically and the simplified form.

Original expression: (1 + √2)/(2 + √3). Compute each part:

√2 ≈ 1.4142, √3 ≈ 1.7320.

Numerator: 1 + 1.4142 ≈ 2.4142

Denominator: 2 + 1.7320 ≈ 3.7320

So original expression ≈ 2.4142 / 3.7320 ≈ 0.6455

Simplified form: 2 - √3 + 2√2 - √6 ≈ 2 - 1.7320 + 2*1.4142 - 2.4495 ≈ 2 - 1.7320 + 2.8284 - 2.4495 ≈ (2 - 1.7320) + (2.8284 - 2.4495) ≈ 0.2680 + 0.3789 ≈ 0.6469. Hmm, close to 0.6455, but there's a slight difference. Wait, maybe due to rounding errors? Let me compute more precisely.

Compute each term with more decimal places:

√2 ≈ 1.41421356

√3 ≈ 1.73205081

√6 ≈ 2.44948974

Compute numerator: 1 + √2 ≈ 1 + 1.41421356 = 2.41421356

Denominator: 2 + √3 ≈ 2 + 1.73205081 = 3.73205081

Original expression ≈ 2.41421356 / 3.73205081 ≈ Let's compute this division:

2.41421356 ÷ 3.73205081 ≈ 0.645497224

Now the simplified expression:

2 - √3 + 2√2 - √6

Compute each term:

2 = 2.0

-√3 ≈ -1.73205081

2√2 ≈ 2 * 1.41421356 ≈ 2.82842712

-√6 ≈ -2.44948974

Sum all terms:

2.0 - 1.73205081 + 2.82842712 - 2.44948974

Calculate step by step:

2.0 - 1.73205081 = 0.26794919

0.26794919 + 2.82842712 = 3.09637631

3.09637631 - 2.44948974 ≈ 0.64688657

Comparing to original: 0.645497224 vs 0.64688657. The difference is about 0.001389, which is likely due to rounding errors in the decimal approximations. The exact values should be equal. Let me check algebraically.

Original expression: (1 + √2)/(2 + √3). Multiply numerator and denominator by (2 - √3):

[(1 + √2)(2 - √3)] / [(2 + √3)(2 - √3)] = [2 - √3 + 2√2 - √6] / 1 = 2 - √3 + 2√2 - √6

Which is exactly the simplified form. Therefore, the decimal approximations are slightly different due to rounding, but the expressions are equal. Therefore, the simplification is correct.

Therefore, the expression simplifies to 2(1 + √2) - (√3 + √6), so A=2, B=2, C=3, D=6. Thus, A + B + C + D = 2 + 2 + 3 + 6 = 13.

I think that's the answer. Let me just check once more if there's another way to express this. Suppose someone factors differently. For example, could it be written as (2 + 2√2) - (√3 + √6)? But that's not in the form A(1 + √B). Alternatively, if someone tried to factor out something else, but I don't see another way. So I think the answer is 13.

**Final Answer**
\boxed{13}
Thus, the sum $A + B + C + D$ is $\boxed{13}$.
</think>