`(sqrt3+1)^2=?+sqrt12`

Options

A. `2sqrt(3)+4`
B. `sqrt3`
C. `sqrt2`
D. 4

Answer

To solve the equation \((\sqrt{3} + 1)^2 = ? + \sqrt{12}\), we will follow these steps:

### Step 1: Expand the left side of the equation
Using the formula for squaring a binomial, \((a + b)^2 = a^2 + 2ab + b^2\), we can expand \((\sqrt{3} + 1)^2\).

\[ (\sqrt{3} + 1)^2 = (\sqrt{3})^2 + 2(\sqrt{3})(1) + (1)^2 \]

### Step 2: Calculate each term
Now we calculate each term in the expansion:

- \((\sqrt{3})^2 = 3\)
- \(2(\sqrt{3})(1) = 2\sqrt{3}\)
- \((1)^2 = 1\)

Putting it all together:

\[ (\sqrt{3} + 1)^2 = 3 + 2\sqrt{3} + 1 \]

### Step 3: Combine like terms
Now we combine the constant terms:

\[ 3 + 1 = 4 \]

So, we have:

\[ (\sqrt{3} + 1)^2 = 4 + 2\sqrt{3} \]

### Step 4: Set the equation
Now we set the expanded left side equal to the right side of the original equation:

\[ 4 + 2\sqrt{3} = ? + \sqrt{12} \]

### Step 5: Simplify \(\sqrt{12}\)
We can simplify \(\sqrt{12}\):

\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \]

### Step 6: Substitute back into the equation
Now we substitute \(\sqrt{12}\) back into the equation:

\[ 4 + 2\sqrt{3} = ? + 2\sqrt{3} \]

### Step 7: Isolate the question mark
To isolate the question mark, we subtract \(2\sqrt{3}\) from both sides:

\[ 4 + 2\sqrt{3} - 2\sqrt{3} = ? \]

This simplifies to:

\[ 4 = ? \]

### Final Answer
Thus, the value of \(?\) is:

\[ ? = 4 \]