The coefficient of x^5 in (3x - 4)^7 is C(7, 5) * (3)^5 * (-4)^2 = 21 * 243 * 16 = 68016.

Using the binomial theorem, the coefficient of x^2 in (2x - 3)^4 is given by 4C2 * (2)^2 * (-3)^2 = 6 * 4 * 9 = 216.

The coefficient of x^2 in (3x - 2)^5 is given by 5C2 * (3x)^2 * (-2)^3 = 10 * 9 * (-8) = -720.

The coefficient of x^4 is C(8, 4) = 70.

(1 + i)² = 1² + 2(1)(i) + i² = 1 + 2i - 1 = 2i.

The coefficient of x^2 in (x + k)^4 is C(4, 2) * k^2 = 6. Thus, 6k^2 = 6, giving k^2 = 1, so k = 1 or -1.

The term containing x^4 is C(6,4) * (2)^2 * x^4 = 15 * 4 * x^4 = 60x^4. Setting 60 = 240 gives k = 4.

C(7, 4) = 7! / (4! * (7-4)!) = 7! / (4! * 3!) = (7*6*5)/(3*2*1) = 35.

Using the binomial theorem, the coefficient of x^4 in (a + b)^n is given by nCk * a^(n-k) * b^k. Here, n=6, a=x, b=-2, and k=2. Thus, the coefficient is 6C2 * (1)^4 * (-2)^2 = 15 * 4 = 60.

By substituting x = 1 into the equation, we find that it satisfies the equation, hence 1 is a root.

The condition for no real roots is that the discriminant must be less than zero: 6^2 - 4*1*k < 0 => 36 < 4k => k > 9.

The equation can be factored as (x-1)^3 = 0, which has one real root (x = 1) with multiplicity 3.

The product of the roots of the cubic equation ax^3 + bx^2 + cx + d = 0 is given by -d/a. Here, d = -6 and a = 1, so the product is -(-6)/1 = 6.

By substituting x = 2 into the equation, we find that 2 is a root since 2^3 - 6(2^2) + 11(2) - 6 = 0.

The polynomial can be factored as (x-1)^3, indicating that it has one real root with multiplicity 3, hence all roots are real and equal.

The sum of the roots is given by -b/a = 3/1 = 3.

The polynomial can be factored as (x - 1)^3, indicating that all roots are real and equal.

The discriminant D = b^2 - 4ac = 4^2 - 4(2)(2) = 16 - 16 = 0.

For equal roots, the discriminant must be zero: b^2 - 4ac = 0. Here, 4^2 - 4(2)(k) = 0 leads to k = 4.

The discriminant must be non-negative: 4^2 - 4*2*k >= 0, which simplifies to k <= 4.

The discriminant D = b^2 - 4ac = 4^2 - 4(2)(-6) = 16 + 48 = 64.

For equal roots, the discriminant must be zero: (-4)^2 - 4*2*k = 0. Solving gives k = 4.

The discriminant must be non-negative: (2p)^2 - 4(1)(p^2 - 4) >= 0 leads to p >= 2.

The discriminant D = 6^2 - 4*1*9 = 0, indicating real and equal roots.

The discriminant must be non-negative: 6^2 - 4(1)(k) ≥ 0, which gives k ≤ 9.

The sum of the roots is -(-2) + -(-3) = 5, hence p = 5.

The discriminant is 0, indicating that the roots are real and equal.

If one root is 3, then substituting x = 3 gives 3^2 - 6*3 + k = 0, leading to k = 9.

For equal roots, the discriminant must be zero: k^2 - 4*1*16 = 0. Solving gives k = -8.

The number of ways to choose 3 people from 8 is given by 8C3 = 56.