`2x^(2)+11x+14` =______

Options

A. `(2x+2)(x+7)`
B. `(2x+7)(x+4)`
C. `(2x+7)(x+2)`
D. `(2x+4)(x+7)`

Answer

To factor the quadratic expression \(2x^2 + 11x + 14\), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is in the form \(ax^2 + bx + c\). Here, we have:
- \(a = 2\)
- \(b = 11\)
- \(c = 14\)

### Step 2: Calculate the product \(ac\)
We need to find the product of \(a\) and \(c\):
\[ ac = 2 \times 14 = 28 \]

### Step 3: Find two numbers that multiply to \(ac\) and add to \(b\)
We need to find two numbers that multiply to \(28\) (the value of \(ac\)) and add up to \(11\) (the value of \(b\)). The pairs of factors of \(28\) are:
- \(1 \times 28\)
- \(2 \times 14\)
- \(4 \times 7\)

Among these, the pair \(4\) and \(7\) adds up to \(11\):
\[ 4 + 7 = 11 \]

### Step 4: Rewrite the middle term
Now, we can rewrite the expression \(2x^2 + 11x + 14\) by splitting the middle term using the numbers we found:
\[ 2x^2 + 4x + 7x + 14 \]

### Step 5: Factor by grouping
Next, we group the terms:
\[ (2x^2 + 4x) + (7x + 14) \]
Now, factor out the common factors from each group:
\[ 2x(x + 2) + 7(x + 2) \]

### Step 6: Factor out the common binomial factor
Now, we can factor out the common binomial factor \((x + 2)\):
\[ (2x + 7)(x + 2) \]

### Final Answer
Thus, the factored form of the expression \(2x^2 + 11x + 14\) is:
\[ (2x + 7)(x + 2) \]

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