Question

factor completely. 21d² + 77d + 56

Explanation:

Step1: Find GCD of coefficients

Find GCD of 21, 77, 56. GCD(21,77,56) = 7.
$$21d^2 + 77d + 56 = 7(3d^2 + 11d + 8)$$

Step2: Factor quadratic

Factor \(3d^2 + 11d + 8\). Find two numbers: \(3\times8 = 24\), factors of 24 that sum to 11: 3 and 8.
$$3d^2 + 3d + 8d + 8 = 3d(d + 1) + 8(d + 1) = (3d + 8)(d + 1)$$

Step3: Combine factors

Combine with GCD factor: \(7(3d + 8)(d + 1)\)

Answer:

\(7(3d + 8)(d + 1)\)