To factor the trinomial \(3x^2 + 22x + 7\), we will look for two numbers that multiply to \(3 \times 7 = 21\) and add to \(22\).
The numbers that work are \(21\) and \(1\).
Now we can rewrite the middle term using these two numbers: \[ 3x^2 + 21x + 1x + 7 \]
Next, we can group the terms: \[ (3x^2 + 21x) + (1x + 7) \]
Now, we factor out the common factors in each group: \[ 3x(x + 7) + 1(x + 7) \]
Next, we can factor out the common binomial \((x + 7)\): \[ (3x + 1)(x + 7) \]
Thus, the expression equivalent to the trinomial \(3x^2 + 22x + 7\) is:
\[ (3x + 1)(x + 7) \]
In the requested format with parentheses around each factor, your answer is: \[ (3x + 1)(x + 7) \]