To simplify the expression \((3 \sqrt{5} + \sqrt{3}) (3 \sqrt{5} + \sqrt{3})\), we can use the formula for squaring a binomial, \((a + b)^2 = a^2 + 2ab + b^2\).

Here, \(a = 3 \sqrt{5}\) and \(b = \sqrt{3}\).

### Step-by-step solution:
\[
(3 \sqrt{5} + \sqrt{3}) (3 \sqrt{5} + \sqrt{3}) = (3 \sqrt{5})^2 + 2(3 \sqrt{5})(\sqrt{3}) + (\sqrt{3})^2
\]

1. First term: \((3 \sqrt{5})^2 = 9 \times 5 = 45\)
2. Second term: \(2(3 \sqrt{5})(\sqrt{3}) = 6 \sqrt{15}\)
3. Third term: \((\sqrt{3})^2 = 3\)

Now, add the results:
\[
45 + 6 \sqrt{15} + 3 = 48 + 6 \sqrt{15}
\]

Thus, the simplified expression is:
\[
\boxed{48 + 6 \sqrt{15}}
\]

Would you like further explanation or have any questions?

### Related questions:
1. How would you simplify \((2 \sqrt{7} - 4)(2 \sqrt{7} - 4)\)?
2. What happens if we multiply \((3 \sqrt{5} + \sqrt{3})\) by \((3 \sqrt{5} - \sqrt{3})\)?
3. Can you expand and simplify \((2 \sqrt{6} + \sqrt{2})(2 \sqrt{6} + \sqrt{2})\)?
4. How do we simplify an expression involving cube roots instead of square roots?
5. What is the geometric interpretation of squaring a binomial involving square roots?

### Tip:
Always remember that \((a + b)^2 = a^2 + 2ab + b^2\) — a handy tool for quickly expanding binomials!

Learn how to simplify the expression (3 √(5) + √(3))(3 √(5) + √(3)) using the binomial expansion formula. This step-by-step solution includes core concepts in algebra and square roots, suitable for students in Grades 9-10.

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