Factorise:
`x ^(2) + 13x + 40`

Answer

To factorise the expression \( x^2 + 13x + 40 \), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is \( x^2 + 13x + 40 \). Here, the coefficients are:
- Coefficient of \( x^2 \) (a) = 1
- Coefficient of \( x \) (b) = 13
- Constant term (c) = 40

### Step 2: Multiply the constant term by the coefficient of \( x^2 \)
We need to multiply the constant term \( c \) (which is 40) by the coefficient of \( x^2 \) (which is 1):
\[ a \cdot c = 1 \cdot 40 = 40 \]

### Step 3: Find two numbers that multiply to 40 and add to 13
Now, we need to find two numbers that multiply to 40 and add up to 13. Let's list the factor pairs of 40:
- \( 1 \times 40 \)
- \( 2 \times 20 \)
- \( 4 \times 10 \)
- \( 5 \times 8 \)

Among these pairs, the pair \( 5 \) and \( 8 \) satisfies our requirement:
\[ 5 + 8 = 13 \quad \text{and} \quad 5 \times 8 = 40 \]

### Step 4: Rewrite the middle term using the two numbers
We can rewrite the expression \( x^2 + 13x + 40 \) by splitting the middle term \( 13x \) into \( 5x + 8x \):
\[ x^2 + 5x + 8x + 40 \]

### Step 5: Factor by grouping
Now, we can group the terms:
\[ (x^2 + 5x) + (8x + 40) \]
Factoring out the common factors in each group:
\[ x(x + 5) + 8(x + 5) \]

### Step 6: Factor out the common binomial
Now we can factor out the common binomial \( (x + 5) \):
\[ (x + 5)(x + 8) \]

### Final Answer
Thus, the factorised form of \( x^2 + 13x + 40 \) is:
\[ (x + 5)(x + 8) \]

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