To factor the quadratic expression \(3x^2 + 22x + 7\), we will look for two binomials in the form \((ax + b)(cx + d)\).
- First, we check the product \(a \cdot c\) which is the coefficient of \(x^2\) (which is 3).
- We also need the product \(b \cdot d\) which is the constant term (which is 7).
- The middle term (22x) should be the result of adding the cross products of the binomials.
We will look for pairs of factors of \(3 \cdot 7 = 21\) that add up to 22.
The pairs of factors of 21 are:
- (1, 21)
- (3, 7)
Now, we can see that \(1\) and \(21\) add up to \(22\). Therefore, we can express 22x as \(1x + 21x\).
Now, we can rewrite the quadratic as follows:
\[ 3x^2 + 1x + 21x + 7 \]
Next, we group the terms:
\[ (3x^2 + 1x) + (21x + 7) \]
Now, we can factor each group:
\[ x(3x + 1) + 7(3x + 1) \]
We can see that \(3x + 1\) is a common factor:
\[ (3x + 1)(x + 7) \]
Thus, the factored form of \(3x^2 + 22x + 7\) is:
\[ (3x + 1)(x + 7) \]