For each of the following expression, find all the values of $k$ so that the trinomial can be factored over the integers 16 A a) $x^{2}+12 x+k$ the vabes olk b) $2 x^{2}+k x+15$ the k of ? nat allow to be factaed $27+k x+15$ values ruat allow tuetr: $27+k x+15$ over integers. c) $(x-1)^{2}-k$ the values that $f_{14 T}$ $(x-k)^{2}$ Allows rue $k^{2}$ $(x-k)^{2}$ to be the factored,

Step 1 ：\(a)\ x^2 + 12x + k = (x + a)(x + b)\)

Step 2 ：\(ab = k\ and \ a + b = 12\)

Step 3 ：\(k = 1 \cdot 11, 2 \cdot 6, 3 \cdot 4, -1 \cdot -11, -2 \cdot -6, -3 \cdot -4\)

Step 4 ：\(k = 11, 12, 6, -11, -12, -6\)

Step 5 ：\(b)\ 2x^2 + kx + 15 = (2x + a)(x + b)\)

Step 6 ：\(2ab = k\ and \ a + 2b = k\)

Step 7 ：\(k = 2 \cdot 1 \cdot 15, 2 \cdot 3 \cdot 5, -2 \cdot -1 \cdot -15, -2 \cdot -3 \cdot -5\)

Step 8 ：\(k = 30, 30, -30, -30\)

Step 9 ：\(c)\ (x - 1)^2 - k = (x - a)(x - b)\)

Step 10 ：\((a - 1)(b - 1) = k\ and \ a + b = 2\)

Step 11 ：\(k = 1 \cdot 1, -1 \cdot 3, -3 \cdot 5, -5 \cdot 7\)

Step 12 ：\(k = 1, -3, -15, -35\)

Step 13 ：\boxed{a: \{11, 12, 6, -11, -12, -6\}, b: \{30, -30\}, c: \{1, -3, -15, -35\}}\)