Factorise the following :
`x^(2)+7x+10`

Answer

To factorise the expression \( x^2 + 7x + 10 \), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is in the form \( ax^2 + bx + c \), where:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = 7 \) (coefficient of \( x \))
- \( c = 10 \) (constant term)

### Step 2: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that:
- Multiply to \( ac = 1 \times 10 = 10 \)
- Add up to \( b = 7 \)

The two numbers that satisfy these conditions are \( 5 \) and \( 2 \), since:
- \( 5 \times 2 = 10 \)
- \( 5 + 2 = 7 \)

### Step 3: Rewrite the middle term using the two numbers
Now we can rewrite the expression by splitting the middle term \( 7x \) into \( 5x + 2x \):
\[ x^2 + 5x + 2x + 10 \]

### Step 4: Group the terms
Next, we group the terms:
\[ (x^2 + 5x) + (2x + 10) \]

### Step 5: Factor out the common factors from each group
From the first group \( (x^2 + 5x) \), we can factor out \( x \):
\[ x(x + 5) \]
From the second group \( (2x + 10) \), we can factor out \( 2 \):
\[ 2(x + 5) \]

### Step 6: Combine the factored terms
Now we can combine the factored terms:
\[ x(x + 5) + 2(x + 5) \]
We see that \( (x + 5) \) is a common factor:
\[ (x + 5)(x + 2) \]

### Final Answer
Thus, the factorised form of \( x^2 + 7x + 10 \) is:
\[ (x + 5)(x + 2) \]

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