Factorise:
`x^(2) + 8x +15`

Options

A. `=(x-5) (x-3)`
B. `=(x+5) (x-3)`
C. `=(x-5) (x+3)`
D. `=(x+5) (x+3)`

Answer

To factorise the quadratic expression \(x^2 + 8x + 15\), we will follow these steps:

### Step 1: Identify the coefficients
The given expression is in the form \(ax^2 + bx + c\), where:
- \(a = 1\) (coefficient of \(x^2\))
- \(b = 8\) (coefficient of \(x\))
- \(c = 15\) (constant term)

### Step 2: Find two numbers that multiply to \(ac\) and add to \(b\)
We need to find two numbers that:
- Multiply to \(ac = 1 \times 15 = 15\)
- Add up to \(b = 8\)

The two numbers that satisfy these conditions are \(3\) and \(5\) because:
- \(3 \times 5 = 15\)
- \(3 + 5 = 8\)

### Step 3: Rewrite the middle term using the two numbers
We can rewrite the expression \(x^2 + 8x + 15\) as:
\[ x^2 + 3x + 5x + 15 \]

### Step 4: Factor by grouping
Now, we will group the terms:
\[ (x^2 + 3x) + (5x + 15) \]

Next, we factor out the common factors from each group:
\[ x(x + 3) + 5(x + 3) \]

### Step 5: Factor out the common binomial
Now, we can factor out the common binomial \((x + 3)\):
\[ (x + 3)(x + 5) \]

### Final Result
Thus, the factorised form of the expression \(x^2 + 8x + 15\) is:
\[ (x + 3)(x + 5) \]