What should come in place of the question mark (?) in the following questions:
`(1+sqrt5)^2 + ? = sqrt20`

Options

A. `6+4sqrt5`
B. -6
C. 6
D. `sqrt6`

Answer

To solve the equation \( (1+\sqrt{5})^2 + ? = \sqrt{20} \), we will follow these steps:

### Step 1: Expand \( (1+\sqrt{5})^2 \)
Using the formula for the square of a binomial, \( (a+b)^2 = a^2 + 2ab + b^2 \), where \( a = 1 \) and \( b = \sqrt{5} \):

\[ (1+\sqrt{5})^2 = 1^2 + 2 \cdot 1 \cdot \sqrt{5} + (\sqrt{5})^2 \]

Calculating each term:
- \( 1^2 = 1 \)
- \( 2 \cdot 1 \cdot \sqrt{5} = 2\sqrt{5} \)
- \( (\sqrt{5})^2 = 5 \)

Putting it all together:
\[ (1+\sqrt{5})^2 = 1 + 2\sqrt{5} + 5 = 6 + 2\sqrt{5} \]

### Step 2: Substitute back into the equation
Now we substitute back into the original equation:

\[ 6 + 2\sqrt{5} + ? = \sqrt{20} \]

### Step 3: Simplify \( \sqrt{20} \)
We can simplify \( \sqrt{20} \):

\[ \sqrt{20} = \sqrt{4 \cdot 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5} \]

### Step 4: Set up the equation to solve for ?
Now we have:

\[ 6 + 2\sqrt{5} + ? = 2\sqrt{5} \]

### Step 5: Isolate the question mark (?)
To find \( ? \), we subtract \( 6 + 2\sqrt{5} \) from both sides:

\[ ? = 2\sqrt{5} - (6 + 2\sqrt{5}) \]

This simplifies to:

\[ ? = 2\sqrt{5} - 6 - 2\sqrt{5} \]

The \( 2\sqrt{5} \) terms cancel out:

\[ ? = -6 \]

### Final Answer
Thus, the value that should come in place of the question mark is:

\[ \boxed{-6} \]