Factorise:
`2x ^(2) -17 x -30`

Answer

To factorise the expression \(2x^2 - 17x - 30\), we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is:
\[ 2x^2 - 17x - 30 \]
Here, the coefficient of \(x^2\) (let's call it \(a\)) is 2, the coefficient of \(x\) (let's call it \(b\)) is -17, and the constant term (let's call it \(c\)) is -30.

### Step 2: Multiply \(a\) and \(c\)
We need to multiply the coefficient of \(x^2\) (which is \(a = 2\)) by the constant term \(c = -30\):
\[ a \cdot c = 2 \cdot (-30) = -60 \]

### Step 3: Find two numbers that multiply to \(-60\) and add to \(-17\)
We need to find two numbers that multiply to \(-60\) and add up to \(-17\). The numbers that satisfy these conditions are \(-20\) and \(3\):
\[ -20 \cdot 3 = -60 \quad \text{and} \quad -20 + 3 = -17 \]

### Step 4: Rewrite the middle term
Now, we can rewrite the expression by splitting the middle term \(-17x\) into \(-20x + 3x\):
\[ 2x^2 - 20x + 3x - 30 \]

### Step 5: Group the terms
Next, we group the terms:
\[ (2x^2 - 20x) + (3x - 30) \]

### Step 6: Factor out the common factors
Now, we factor out the common factors from each group:
1. From \(2x^2 - 20x\), we can factor out \(2x\):
\[ 2x(x - 10) \]
2. From \(3x - 30\), we can factor out \(3\):
\[ 3(x - 10) \]

So, we can rewrite the expression as:
\[ 2x(x - 10) + 3(x - 10) \]

### Step 7: Factor out the common binomial
Now, we see that \((x - 10)\) is a common factor:
\[ (2x + 3)(x - 10) \]

### Final Answer
Thus, the factorised form of \(2x^2 - 17x - 30\) is:
\[ (2x + 3)(x - 10) \]

---