Setting x^2 = 2x + 1 at x = c gives c = 2.

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

The discriminant must be less than zero: k^2 - 4*1*16 < 0 => k^2 < 64 => k < 8 and k > -8.

The discriminant must be positive: k^2 - 4*1*16 > 0 => k^2 > 64 => k > 8 or k < -8. Thus, k = -4 is valid.

Substituting x = 2 into the equation gives 2^2 + 2k + 4 = 0, leading to k = -4.

Let the roots be r and r+2. Then, r + (r+2) = 4 and r(r+2) = k leads to k = 4.

For both roots to be positive, k must be greater than 6, as the sum of the roots must be positive.

The discriminant must be less than zero: k² - 4*1*9 < 0, thus k² < 36, so k > 6 or k < -6.

For both roots to be positive, k must be greater than 0 and k² > 4*9 = 36, thus k > 6.

The discriminant must be negative: k^2 - 16 < 0, hence k > 4.

f'(x) = 2kx + 2; f'(-1) = -2k + 2 must exist for any k.

To ensure continuity at x = 2, k(2) + 1 must equal 3. Thus, k = 1.

For both roots to be negative, m must be greater than 0 and m² < 36. Thus, m must be in the range (-6, 0). The suitable value is -4.

The sum of the roots is 3 + 3 = 6, hence m = 6.

Setting 3 + m = 2 and 2 = m + 1 gives m = 1 for continuity.

Using the sum of roots: 2 + (-2) = -p, hence p = 0.

For both roots to be negative, p must be greater than 0 and p² > 16. Thus, p < -4.

For both roots to be negative, p must be positive and p² > 4*9. Thus, p > 6, so p = -4 is valid.

The sum of the roots is 3 + 3 = 6, hence p = 6.

Setting the two pieces equal at x = 0 gives us p = -1.

The number of ways to choose 4 people from 10 is given by 10C4 = 210.

The number of ways to choose 4 people from 8 is given by 8C4 = 70.

Using tan(45°) = height/distance, we have 1 = 20/distance. Therefore, distance = 20 m.

Using tan(30°) = height/distance, we have 1/√3 = 50/distance. Therefore, distance = 50√3 ≈ 86.6 m.

Using tan(30°) = height/100, we have 1/√3 = height/100. Therefore, height = 100/√3 ≈ 57.74 m.

Using tan(45°) = height/distance, we have distance = height/tan(45°) = 40/1 = 40 m.

Using tan(45°) = height/distance, we have distance = height/tan(45°) = 75/1 = 75 m.

Distance = height / tan(angle) = 50 / 1 = 50 m.

Using tan(30°) = height/distance, we have height = distance * tan(30°) = 100 * (1/√3) = 100/√3 = 50 m.

Using tan(45°) = height/distance, we have distance = height/tan(45°) = 100/1 = 100 m.