The expression in the image is:

$(4a + \sqrt{7})(2a - \sqrt{2})$

To simplify this, we expand the terms using the distributive property (also called the FOIL method for binomials):

$(4a + \sqrt{7})(2a - \sqrt{2})$

Step-by-Step Expansion:

1. Multiply the first terms: $4a \cdot 2a = 8a^2$
2. Multiply the outer terms: $4a \cdot (-\sqrt{2}) = -4a\sqrt{2}$
3. Multiply the inner terms: $\sqrt{7} \cdot 2a = 2a\sqrt{7}$
4. Multiply the last terms: $\sqrt{7} \cdot (-\sqrt{2}) = -\sqrt{14}$

Now, combine these results:

$8a^2 - 4a\sqrt{2} + 2a\sqrt{7} - \sqrt{14}$

Final Simplified Expression:

$8a^2 - 4a\sqrt{2} + 2a\sqrt{7} - \sqrt{14}$

Would you like further explanation or help with similar problems?

Related Questions:

1. How is the FOIL method applied to simplify binomials like this?
2. What are the properties of square roots that allow multiplication, e.g., $\sqrt{7} \cdot \sqrt{2} = \sqrt{14}$?
3. Can this result be factored further?
4. How does this expansion relate to real-world applications?
5. What is the significance of leaving results in terms of square roots?

Tip:

When dealing with square roots in expressions, simplify as much as possible but avoid approximating unless explicitly required, to retain precision.