The given expression is:

$7\sqrt{5} - \sqrt{75} + \sqrt{48} - \sqrt{45}$

Let's simplify each square root term:

1. $\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}$
2. $\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$
3. $\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}$

Now, substitute these values into the original expression:

$7\sqrt{5} - 5\sqrt{3} + 4\sqrt{3} - 3\sqrt{5}$

Group the terms with $\sqrt{5}$ and $\sqrt{3}$:

$(7\sqrt{5} - 3\sqrt{5}) + (-5\sqrt{3} + 4\sqrt{3})$

Simplify each group:

$4\sqrt{5} - \sqrt{3}$

Thus, the simplified form is $4\sqrt{5} - \sqrt{3}$, which corresponds to option a.

Would you like more details on any part of this solution? Here are some follow-up questions:

1. How do you simplify square root terms with factors?
2. What are the properties of square roots in algebraic simplifications?
3. How does grouping terms help in simplifying expressions?
4. What are some common square roots to memorize for simplifications?
5. How can you verify the final result of an expression simplification?

Tip: Always break down square roots into their prime factors to simplify them effectively!