Setting the two pieces equal at x = 1 gives 1 - a = 1. Thus, a = 0.

Setting the two pieces equal at x = 1 gives 1 = a(1) + 1, leading to a = 0.

Setting the left limit (2(1) + 1 = 3) equal to the right limit (1^2 + 1 = 2), we find b = 3.

Setting limit as x approaches 1 gives b = 2 for continuity.

Setting the two pieces equal at x = 2 gives us 0 = 2b + 2. Solving for b gives b = -1.

Setting 1 - 3 + b = 2 gives b = 4 for continuity.

Setting 3(1) + c = 2(1)^2 gives c = -1 for continuity.

Setting the two pieces equal at x = c: c^2 - 4 = 3c - 5. Solving gives c = 3.

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

To make f(x) continuous at x = 1, we need c = 1^2 = 1. Thus, c must be 1.

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.

The function f(x) = kx + 2 is a linear function and is continuous for all k at x = 3.

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

To be continuous at x = 2, k(2) + 1 must equal 3. Thus, 2k + 1 = 3, leading to k = 1.

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

To be continuous at x = 2, k(2) + 1 must equal 3. Thus, 2k + 1 = 3, giving 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.