factorize the given expression`2x^2-7x-15`

Answer

To factorize the expression \(2x^2 - 7x - 15\), we will use the middle term splitting method. Here are the steps:

### Step 1: Identify the coefficients
The given expression is:
\[ 2x^2 - 7x - 15 \]
Here, the coefficient of \(x^2\) is \(2\), the coefficient of \(x\) is \(-7\), and the constant term is \(-15\).

### Step 2: Multiply the coefficient of \(x^2\) by the constant term
We need to multiply the coefficient of \(x^2\) (which is \(2\)) by the constant term (which is \(-15\)):
\[ 2 \times (-15) = -30 \]

### Step 3: Find two numbers that multiply to \(-30\) and add to \(-7\)
We need to find two numbers that multiply to \(-30\) and add up to \(-7\). The factors of \(-30\) that satisfy this condition are \(3\) and \(-10\):
\[ 3 \times (-10) = -30 \quad \text{and} \quad 3 + (-10) = -7 \]

### Step 4: Rewrite the middle term using the two numbers
Now we can rewrite the expression by splitting the middle term \(-7x\) into \(3x\) and \(-10x\):
\[ 2x^2 + 3x - 10x - 15 \]

### Step 5: Group the terms
Next, we group the terms:
\[ (2x^2 + 3x) + (-10x - 15) \]

### Step 6: Factor by grouping
Now we factor each group:
\[ x(2x + 3) - 5(2x + 3) \]

### Step 7: Factor out the common binomial
Now we can factor out the common binomial \((2x + 3)\):
\[ (2x + 3)(x - 5) \]

### Final Answer
Thus, the factorization of the expression \(2x^2 - 7x - 15\) is:
\[ (2x + 3)(x - 5) \]

---