Final answer:
To factor the expression by grouping, we first arrange the terms into pairs with common factors, factor those out, and then factor out the common binomial. After factoring, we get (5d - 3)(2d + 7) and verify that this is correct by expanding and checking against the original expression.
Step-by-step explanation:
To factor the expression 10d² - 6d + 35d - 21 by grouping, we need to rearrange the terms to group them into pairs that share a common factor. Rearranging gives us 10d² + 35d - 6d - 21. Now, we can factor out the common factors from each pair:
- From the first pair, 10d² + 35d, we factor out 5d, resulting in 5d(2d + 7).
- From the second pair, -6d - 21, we factor out -3, resulting in -3(2d + 7).
Both groups now share the common factor (2d + 7), so we can factor that out:
(5d - 3)(2d + 7)
Check the answer to see if it is reasonable by expanding the factored form to see if it equals the original expression:
5d(2d) + 5d(7) - 3(2d) - 3(7) = 10d² + 35d - 6d - 21
This confirms that our factoring by grouping is correct.