Factorise:
`2x^(2) + 9x + 10`

Answer

To factorise the quadratic expression \(2x^2 + 9x + 10\), we can follow these steps:

### Step 1: Identify the coefficients
The quadratic expression is in the form \(ax^2 + bx + c\), where:
- \(a = 2\)
- \(b = 9\)
- \(c = 10\)

### Step 2: Calculate the product of \(a\) and \(c\)
We need to find two numbers that multiply to \(a \cdot c\) (which is \(2 \cdot 10 = 20\)) and add up to \(b\) (which is \(9\)).

### Step 3: Find the two numbers
We need two numbers that multiply to \(20\) and add up to \(9\). The numbers \(4\) and \(5\) satisfy this condition:
- \(4 \cdot 5 = 20\)
- \(4 + 5 = 9\)

### Step 4: Rewrite the middle term
Now, we can rewrite the expression \(2x^2 + 9x + 10\) by splitting the middle term \(9x\) into \(4x + 5x\):
\[ 2x^2 + 4x + 5x + 10 \]

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

### Step 6: Factor out the common factors from each group
From the first group \(2x^2 + 4x\), we can factor out \(2x\):
\[ 2x(x + 2) \]
From the second group \(5x + 10\), we can factor out \(5\):
\[ 5(x + 2) \]

### Step 7: Combine the factored groups
Now we have:
\[ 2x(x + 2) + 5(x + 2) \]
We can see that \((x + 2)\) is a common factor:
\[ (2x + 5)(x + 2) \]

### Final Answer
Thus, the factorised form of \(2x^2 + 9x + 10\) is:
\[ (2x + 5)(x + 2) \]