Let's calculate the expression:

$(3 + \sqrt{7})(5 + 3\sqrt{28})$

Step 1: Simplify $\sqrt{28}$

First, simplify $\sqrt{28}$. Notice that $28 = 4 \times 7$, so:

$\sqrt{28} = \sqrt{4 \times 7} = \sqrt{4} \times \sqrt{7} = 2\sqrt{7}$

Now, the expression becomes:

$(3 + \sqrt{7})(5 + 6\sqrt{7})$

Step 2: Apply the distributive property

Use the distributive property to expand the expression:

$(3 + \sqrt{7})(5 + 6\sqrt{7}) = 3(5 + 6\sqrt{7}) + \sqrt{7}(5 + 6\sqrt{7})$

Step 3: Multiply each term

Expand each part:

1. $3(5 + 6\sqrt{7}) = 3 \times 5 + 3 \times 6\sqrt{7} = 15 + 18\sqrt{7}$
2. $\sqrt{7}(5 + 6\sqrt{7}) = \sqrt{7} \times 5 + \sqrt{7} \times 6\sqrt{7} = 5\sqrt{7} + 6 \times 7 = 5\sqrt{7} + 42$

Step 4: Combine like terms

Now, combine the results from the two expansions:

$15 + 18\sqrt{7} + 5\sqrt{7} + 42$

Combine the constants and the terms with $\sqrt{7}$:

$(15 + 42) + (18\sqrt{7} + 5\sqrt{7}) = 57 + 23\sqrt{7}$

Final Answer:

The simplified expression is:

$\boxed{57 + 23\sqrt{7}}$

Would you like further clarification on any steps? Here are some related questions:

1. How do you simplify radical expressions like $\sqrt{28}$?
2. Can you explain the distributive property in algebra?
3. What are the rules for multiplying square roots?
4. How do you combine like terms in expressions involving radicals?
5. What are other examples of expressions involving radicals and integers?

Tip: When simplifying square roots, always look for perfect squares as factors to simplify them effectively!