`x^2+2sqrt(3)x-24`

Explanation

To factor the polynomial \( x^2 + 2\sqrt{3}x - 24 \), we will follow these steps:

### Step 1: Identify the coefficients
The polynomial can be expressed in the standard form \( ax^2 + bx + c \), where:
- \( a = 1 \)
- \( b = 2\sqrt{3} \)
- \( c = -24 \)

### Step 2: Calculate the product \( ac \)
We need to calculate \( ac \):
\[ ac = 1 \times (-24) = -24 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers \( m \) and \( n \) such that:
- \( m \cdot n = -24 \)
- \( m + n = 2\sqrt{3} \)

After testing various pairs of factors of \(-24\), we find:
- \( m = 4\sqrt{3} \)
- \( n = -2\sqrt{3} \)

### Step 4: Rewrite the middle term using \( m \) and \( n \)
We can rewrite the polynomial by splitting the middle term:
\[ x^2 + 4\sqrt{3}x - 2\sqrt{3}x - 24 \]

### Step 5: Group the terms
Now, we group the terms:
\[ (x^2 + 4\sqrt{3}x) + (-2\sqrt{3}x - 24) \]

### Step 6: Factor by grouping
Now, we factor out the common factors from each group:
1. From the first group \( x^2 + 4\sqrt{3}x \):
\[ x(x + 4\sqrt{3}) \]
2. From the second group \( -2\sqrt{3}x - 24 \):
\[ -2\sqrt{3}(x + 4\sqrt{3}) \]

Now, we can combine these:
\[ x(x + 4\sqrt{3}) - 2\sqrt{3}(x + 4\sqrt{3}) \]

### Step 7: Factor out the common binomial
Now we can factor out the common binomial \( (x + 4\sqrt{3}) \):
\[ (x + 4\sqrt{3})(x - 2\sqrt{3}) \]

### Final Answer
Thus, the factored form of the polynomial \( x^2 + 2\sqrt{3}x - 24 \) is:
\[ (x + 4\sqrt{3})(x - 2\sqrt{3}) \]

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