Factor the following:

(A) $9a^2 + b^2 + 4c^2 - 6ab + 12ac - 4bc$
(B) $16s^2 + 25t^2 - 40st$
(C) $r^2 - r - 42$
(D) $49g^2 + 14gh + h^2$
(E) $64u^2 + 121v^2 + 4w^2 - 176uv - 32uw + 44vw$                                         [Page No. 82]

$$\begin{aligned} 11. \text{ (A) } 9a^2 + b^2 + 4c^2 - 6ab + 12ac - 4bc \\ &= (3a)^2 + (-b)^2 + (2c)^2 + 2(3a)(-b) + 2(-b) \\ &\quad (2c) + 2(3a)(2c) \\ &= (3a - b + 2c)^2 \\ (B) \quad 16s^2 + 25t^2 - 40st &= (4s)^2 - 2(4s)(5t) + (5t)^2 \\ &= (4s - 5t)^2 \\ (C) \quad \text{Use } a + b = -1, ab = -42 \\ \text{So, } a = -7, b = 6 \\ r^2 - r - 42 &= (r - 7)(r + 6) \\ (D) \quad 49g^2 + 14gh + h^2 &= (7g)^2 + 2(7g)(h) + h^2 \\ &= (7g + h)^2 \\ (E) \quad 64u^2 + 121v^2 + 4w^2 - 176uv - 32uw + 44vw \\ &= (8u)^2 + (-11v)^2 + (-2w)^2 + 2(8u)(-11v) \\ &\quad + 2(-11v)(-2w) + 2(8u)(-2w) \\ &= (8u - 11v - 2w)^2 \end{aligned}$$