Explain the steps in using the identity x2+(a+b)x+ab = (x+a)(x+b) to find the factored form of the polynomial x2+10x+21 . Make sure you include the factored form of the polynomial in your answer.

To factor the polynomial \( x^2 + 10x + 21 \) using the identity \( x^2 + (a+b)x + ab = (x+a)(x+b) \), we need to identify the values of \( a \) and \( b \) that satisfy this identity.

Here are the steps to find the factored form:

1. **Identify \( a + b \) and \( ab \)**:
- In the polynomial \( x^2 + 10x + 21 \):
- \( a + b = 10 \)
- \( ab = 21 \)

2. **Find \( a \) and \( b \)**:
- We need to find two numbers \( a \) and \( b \) that satisfy both conditions: their sum is 10, and their product is 21.
- To do this, we'll list pairs of factors of 21:
- \( (1, 21) \)
- \( (3, 7) \)
- Now, we check which pair sums to 10:
- \( 1 + 21 = 22 \) (not a match)
- \( 3 + 7 = 10 \) (this matches)

Thus, we have found \( a = 3 \) and \( b = 7 \).

3. **Use the factors to write the polynomial in factored form**:
- Now that we have \( a = 3 \) and \( b = 7 \), we can use the identity:
\[
x^2 + 10x + 21 = (x + 3)(x + 7)
\]

4. **Final Result**:
Therefore, the factored form of the polynomial \( x^2 + 10x + 21 \) is:
\[
(x + 3)(x + 7)
\]