The expression provided is:

$\left( \sqrt{3} + 7 \right) \left( \sqrt{3} + 7 \right).$

This can be simplified by applying the formula for squaring a binomial:

$(a + b)^2 = a^2 + 2ab + b^2.$

Here, $a = \sqrt{3}$ and $b = 7$. Substituting these values:

$\left( \sqrt{3} + 7 \right)^2 = (\sqrt{3})^2 + 2(\sqrt{3})(7) + 7^2.$

Now calculate each term:

1. $(\sqrt{3})^2 = 3$,
2. $2(\sqrt{3})(7) = 14\sqrt{3}$,
3. $7^2 = 49$.

Adding these together:

$3 + 14\sqrt{3} + 49 = 52 + 14\sqrt{3}.$

Thus, the simplified expression is:

$\boxed{52 + 14\sqrt{3}}$

Would you like me to explain any steps in more detail?
Here are 5 follow-up questions related to this topic:

1. How do you simplify an expression with more complex radicals?
2. What is the general formula for expanding $(a + b)(a + b)$?
3. Can this type of simplification apply to differences, like $(\sqrt{3} - 7)^2$?
4. What happens when you multiply two different binomials, e.g., $(\sqrt{3} + 7)(\sqrt{3} - 7)$?
5. How do you rationalize a denominator with a radical?

Tip: Always check if the resulting terms can be further simplified when working with radicals!