Fill in the blanks to complete the following identities:

(A) $s^2 - 11s + 24 = (\underline{\hspace{2cm}})(\underline{\hspace{2cm}})$

(B) $(\underline{\hspace{2cm}})(x + 1) = 3x^2 - 4x - 7$

(C) $10x^2 - 11x - 6 = (2x - \underline{\hspace{2cm}})(\underline{\hspace{2cm}} + 2)$

(D) $6x^2 + 7x + 2 = (\underline{\hspace{2cm}})(\underline{\hspace{2cm}})$

(A) Use $a + b = -11, ab = 24$

$$\text{So, } a = -3, b = -8$$

$$s^2 - 11s + 24 = s^2 - 3s - 8s + 24$$

$$= (s - 3)(s - 8)$$

(B) Divide $3x^2 - 4x - 7$ by $(x + 1)$:

$$\begin{aligned} 3x^2 - 4x - 7 &= 3x^2 + 3x - 7x - 7 \\ &= 3x(x + 1) - 7(x + 1) \\ &= (3x - 7)(x + 1) \end{aligned}$$

$$(3x - 7)(x + 1) = 3x^2 - 4x - 7$$

(C) Split $-11x$ using factors of $10 \times (-6) = -60$:

$$-15 + 4 = -11 \text{ and } -15 \times 4 = -60$$

$$\begin{aligned} 10x^2 - 11x - 6 &= 10x^2 - 15x + 4x - 6 \\ &= 5x(2x - 3) + 2(2x - 3) \\ &= (2x - 3)(5x + 2) \end{aligned}$$

(D) Split $7x$ using factors of 12:

$$\begin{aligned} 3 + 4 &= 7, 3 \times 4 = 12 \\ 6x^2 + 7x + 2 &= 6x^2 + 3x + 4x + 2 \\ &= 3x(2x + 1) + 2(2x + 1) \\ &= (2x + 1)(3x + 2) \end{aligned}$$