https://file.aiquickdraw.com/mathbot/file/172607985303800ejqx7o.jpg

Let's analyze the equation in the image step by step. 

The equation is:

\[
(\sqrt{7} + 11)^2 = (?)^2 + 2\sqrt{847} + 122
\]

We'll proceed to solve both sides of the equation to determine the correct value for \( ? \).

### Step 1: Expand the left-hand side
\[
(\sqrt{7} + 11)^2 = (\sqrt{7})^2 + 2(\sqrt{7})(11) + 11^2
\]
\[
= 7 + 22\sqrt{7} + 121
\]
\[
= 128 + 22\sqrt{7}
\]

So the left-hand side of the equation is \( 128 + 22\sqrt{7} \).

### Step 2: Simplify the right-hand side
The right-hand side is:
\[
(?)^2 + 2\sqrt{847} + 122
\]
We need to simplify \( 2\sqrt{847} \). Notice that \( 847 = 121 \times 7 \), so:
\[
\sqrt{847} = \sqrt{121 \times 7} = 11\sqrt{7}
\]
Thus:
\[
2\sqrt{847} = 2 \times 11\sqrt{7} = 22\sqrt{7}
\]

The right-hand side becomes:
\[
(?)^2 + 22\sqrt{7} + 122
\]

### Step 3: Compare both sides
We now have the equation:
\[
128 + 22\sqrt{7} = (?)^2 + 22\sqrt{7} + 122
\]
By canceling out the \( 22\sqrt{7} \) terms on both sides, we are left with:
\[
128 = (?)^2 + 122
\]
Subtract 122 from both sides:
\[
6 = (?)^2
\]
Taking the square root of both sides:
\[
? = \sqrt{6}
\]

Thus, the value of \( ? \) is \( \sqrt{6} \).

### Step 4: Choose the correct option
The answer is:

\[
\boxed{(c) \ 6}
\]

---

Would you like further clarification on any part of this process?

Here are 5 related questions:
1. How do you expand binomials like \( (a + b)^2 \)?
2. Why does \( \sqrt{847} \) simplify to \( 11\sqrt{7} \)?
3. What are the rules for simplifying square roots involving products?
4. How can you solve quadratic equations using square roots?
5. What happens when you cancel terms from both sides of an equation?

**Tip:** Always check if terms with square roots can be factored or simplified to make equations easier to solve.

Solve the equation (sqrt(7) + 11)^2 and determine the value for '?' in the equation given.

Detailed explanation of solving quadratic equations involving square roots and exponents. This problem uses algebraic formulas and theorems such as square root properties and quadratic expansion. Suitable for high school students learning advanced algebra topics.

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