Factor the following quadratic expressions.
`x^(2)+14x+33`

Answer

To factor the quadratic expression \( x^2 + 14x + 33 \), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is in the form \( ax^2 + bx + c \), where:
- \( a = 1 \)
- \( b = 14 \)
- \( c = 33 \)

### Step 2: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that:
- Multiply to \( ac = 1 \times 33 = 33 \)
- Add to \( b = 14 \)

The pairs of factors of 33 are:
- \( 1 \times 33 \)
- \( 3 \times 11 \)

Among these pairs, \( 3 \) and \( 11 \) add up to \( 14 \):
- \( 3 + 11 = 14 \)

### Step 3: Rewrite the middle term using the two numbers
We can rewrite the expression by splitting the middle term:
\[ x^2 + 3x + 11x + 33 \]

### Step 4: Factor by grouping
Now, we can group the terms:
\[ (x^2 + 3x) + (11x + 33) \]
Factoring out the common terms in each group:
\[ x(x + 3) + 11(x + 3) \]

### Step 5: Factor out the common binomial
Now we can factor out the common binomial \( (x + 3) \):
\[ (x + 3)(x + 11) \]

### Final Answer
Thus, the factored form of the quadratic expression \( x^2 + 14x + 33 \) is:
\[ (x + 3)(x + 11) \]

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