`9x^(2) +30 x + 25 `= ______

Options

A. `(3x+5)(3x-10)`
B. `(3x+5)(3x+5)`
C. `(3x+12) (3x+13)`
D. `(9x+5)(x+5)`

Answer

To factor the expression \(9x^2 + 30x + 25\), we can follow these steps:

### Step 1: Identify the coefficients
The expression is in the standard quadratic form \(ax^2 + bx + c\), where:
- \(a = 9\)
- \(b = 30\)
- \(c = 25\)

### Step 2: Rewrite the expression
We can rewrite the expression as:
\[ 9x^2 + 30x + 25 \]

### Step 3: Factor out the perfect squares
Notice that \(9x^2\) is a perfect square (it can be written as \((3x)^2\)) and \(25\) is also a perfect square (it can be written as \(5^2\)).

### Step 4: Find the middle term
Next, we need to check if the middle term \(30x\) can be expressed in the form of \(2ab\) where \(a = 3x\) and \(b = 5\):
\[ 2ab = 2 \cdot (3x) \cdot 5 = 30x \]
This confirms that the expression can be factored using the identity \(a^2 + 2ab + b^2 = (a + b)^2\).

### Step 5: Apply the identity
Using the identity, we can factor the expression:
\[ 9x^2 + 30x + 25 = (3x + 5)^2 \]

### Step 6: Write the final answer
Thus, the factored form of the expression is:
\[ (3x + 5)(3x + 5) \quad \text{or} \quad (3x + 5)^2 \]

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