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!