Factorise:
`x^(2)+ 10 x + 25`

Answer

To factorise the expression \( x^2 + 10x + 25 \), we can follow these steps:

### Step 1: Identify the quadratic expression
Recognize that the expression is a quadratic in the form \( ax^2 + bx + c \), where:
- \( a = 1 \)
- \( b = 10 \)
- \( c = 25 \)

### Step 2: Find two numbers that multiply and add to specific values
We need to find two numbers that:
- Multiply to \( c \) (which is 25)
- Add to \( b \) (which is 10)

### Step 3: Determine the two numbers
The two numbers that satisfy these conditions are both \( 5 \):
- \( 5 \times 5 = 25 \) (product)
- \( 5 + 5 = 10 \) (sum)

### Step 4: Rewrite the middle term
We can rewrite the quadratic expression by splitting the middle term \( 10x \) into \( 5x + 5x \):
\[ x^2 + 5x + 5x + 25 \]

### Step 5: Group the terms
Now, we group the terms:
\[ (x^2 + 5x) + (5x + 25) \]

### Step 6: Factor out the common terms
From the first group \( (x^2 + 5x) \), we can factor out \( x \):
\[ x(x + 5) \]
From the second group \( (5x + 25) \), we can factor out \( 5 \):
\[ 5(x + 5) \]

### Step 7: Combine the factors
Now, we can combine the two factored groups:
\[ x(x + 5) + 5(x + 5) = (x + 5)(x + 5) \]

### Step 8: Write the final factored form
Thus, the expression can be written as:
\[ (x + 5)^2 \]

### Final Answer:
The factorised form of \( x^2 + 10x + 25 \) is \( (x + 5)^2 \).

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