Type your answers to each problem in the table below. The order of the factors doesn't matter, but use parentheses around each one!

Question

3x^ + 22x + 7

Answer 1

To factor the quadratic expression \(3x^2 + 22x + 7\), we can use the method of factoring by grouping or the quadratic formula. However, let's first try to factor it directly.

We want to express it in the form of \((ax + b)(cx + d)\).

1. The product of \( a \) and \( c \) should equal the coefficient of \( x^2 \), which is \( 3 \).
2. The product of \( b \) and \( d \) should equal the constant term, which is \( 7 \).
3. The sum \( ac + bd \) should equal \( 22 \).

Let's start with \( a = 3 \) and \( c = 1 \) (since \( 3 \cdot 1 = 3 \)), and see if we can find values for \( b \) and \( d \):

The factors of \( 7 \) are \( (1, 7) \) or \( (7, 1) \).

Let's test \( (3x + 1)(x + 7) \):

• Expand: \[ (3x + 1)(x + 7) = 3x^2 + 21x + 1x + 7 = 3x^2 + 22x + 7 \]

This matches. Thus, the factorization of the quadratic expression is: \[ (3x + 1)(x + 7) \]

In the requested format, the factors are: \[ (3x + 1)(x + 7) \]