Factorise:
`x^(2) + 5x +6`

Answer

To factorise the expression \( x^2 + 5x + 6 \), we can follow these steps:

### Step 1: Identify the coefficients
In the expression \( x^2 + 5x + 6 \):
- The coefficient of \( x^2 \) (which we denote as \( a \)) is 1.
- The coefficient of \( x \) (denote as \( b \)) is 5.
- The constant term (denote as \( c \)) is 6.

### Step 2: Multiply \( a \) and \( c \)
We need to multiply the coefficient of \( x^2 \) (which is 1) by the constant term (which is 6):
\[ a \cdot c = 1 \cdot 6 = 6 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to 6 and add up to 5. The pairs of factors of 6 are:
- \( 1 \times 6 = 6 \) (and \( 1 + 6 = 7 \))
- \( 2 \times 3 = 6 \) (and \( 2 + 3 = 5 \))

The numbers we are looking for are 2 and 3.

### Step 4: Rewrite the middle term
Now we can rewrite the expression \( x^2 + 5x + 6 \) by splitting the middle term using the numbers we found:
\[ x^2 + 2x + 3x + 6 \]

### Step 5: Factor by grouping
Now, we can group the terms:
\[ (x^2 + 2x) + (3x + 6) \]

Now, we factor out the common factors in each group:
- From the first group \( x^2 + 2x \), we can factor out \( x \):
\[ x(x + 2) \]
- From the second group \( 3x + 6 \), we can factor out \( 3 \):
\[ 3(x + 2) \]

Now we have:
\[ x(x + 2) + 3(x + 2) \]

### Step 6: Factor out the common binomial
Now we can factor out the common binomial \( (x + 2) \):
\[ (x + 2)(x + 3) \]

### Final Result
Thus, the factorised form of \( x^2 + 5x + 6 \) is:
\[ (x + 2)(x + 3) \]

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