`6x^2+17x+5=?`

Options

A. `(2x+1)(3x+5)`
B. `(2x+1)(2x-3)`
C. `(6x+5)(x+1)`
D. None of these

Answer

To factor the polynomial \(6x^2 + 17x + 5\), we can follow these steps:

### Step 1: Identify the coefficients
The given polynomial is in the form \(ax^2 + bx + c\), where:
- \(a = 6\)
- \(b = 17\)
- \(c = 5\)

### Step 2: Multiply \(a\) and \(c\)
We need to multiply the coefficient of \(x^2\) (which is \(a\)) and the constant term \(c\):
\[ a \cdot c = 6 \cdot 5 = 30 \]

### Step 3: Find two numbers that multiply to \(ac\) and add to \(b\)
We need to find two numbers that multiply to \(30\) (from step 2) and add up to \(17\) (the coefficient \(b\)). The numbers that satisfy this condition are \(15\) and \(2\) because:
\[ 15 \cdot 2 = 30 \quad \text{and} \quad 15 + 2 = 17 \]

### Step 4: Rewrite the middle term
We can now rewrite the polynomial by splitting the middle term \(17x\) into \(15x\) and \(2x\):
\[ 6x^2 + 15x + 2x + 5 \]

### Step 5: Group the terms
Next, we group the terms into two pairs:
\[ (6x^2 + 15x) + (2x + 5) \]

### Step 6: Factor out the common factors in each group
Now, we factor out the common factors from each group:
- From the first group \(6x^2 + 15x\), we can factor out \(3x\):
\[ 3x(2x + 5) \]
- From the second group \(2x + 5\), we can factor out \(1\):
\[ 1(2x + 5) \]

### Step 7: Combine the factored terms
Now we can combine the factored terms:
\[ 3x(2x + 5) + 1(2x + 5) \]
We see that \(2x + 5\) is common in both terms, so we can factor it out:
\[ (2x + 5)(3x + 1) \]

### Final Answer
Thus, the factorization of the polynomial \(6x^2 + 17x + 5\) is:
\[ (2x + 5)(3x + 1) \]

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