`x^2+5sqrt(5)x+30`

Explanation

To factor the polynomial \( x^2 + 5\sqrt{5}x + 30 \), we can follow these steps:

### Step 1: Identify coefficients
The polynomial is in the standard form \( ax^2 + bx + c \). Here, we have:
- \( a = 1 \)
- \( b = 5\sqrt{5} \)
- \( c = 30 \)

### Step 2: Calculate the discriminant
The discriminant \( D \) is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (5\sqrt{5})^2 - 4 \cdot 1 \cdot 30 \]
Calculating \( (5\sqrt{5})^2 \):
\[ (5\sqrt{5})^2 = 25 \cdot 5 = 125 \]
Now substituting this back into the discriminant:
\[ D = 125 - 120 = 5 \]
Since \( D > 0 \), the roots are real and distinct.

### Step 3: Factor the quadratic
We need to express the middle term \( 5\sqrt{5}x \) in terms of two numbers that multiply to \( ac = 30 \) and add to \( b = 5\sqrt{5} \).

We can express \( 30 \) as:
\[ 30 = 3 \cdot 10 \]
Next, we can express \( 10 \) in terms of square roots:
\[ 10 = 2\sqrt{5} \cdot \sqrt{5} \]
Now we can rewrite \( 30 \) as:
\[ 30 = 3\sqrt{5} \cdot 2\sqrt{5} \]
This gives us the two numbers \( 3\sqrt{5} \) and \( 2\sqrt{5} \).

### Step 4: Rewrite the polynomial
Now we can rewrite the polynomial:
\[ x^2 + 3\sqrt{5}x + 2\sqrt{5}x + 30 \]
Grouping the terms:
\[ = (x^2 + 3\sqrt{5}x) + (2\sqrt{5}x + 30) \]

### Step 5: Factor by grouping
Now we can factor by grouping:
\[ = x(x + 3\sqrt{5}) + 2\sqrt{5}(x + 3\sqrt{5}) \]
Now we can factor out the common term \( (x + 3\sqrt{5}) \):
\[ = (x + 3\sqrt{5})(x + 2\sqrt{5}) \]

### Final Answer
Thus, the factorization of the polynomial \( x^2 + 5\sqrt{5}x + 30 \) is:
\[ (x + 3\sqrt{5})(x + 2\sqrt{5}) \]