To find the points of intersection, set the equations equal: \(2x + 3 = cx^2 + 3x - c\).

Rearrange to form a quadratic equation: \(cx^2 + (3 - 2c)x - (c + 3) = 0\).

The discriminant of a quadratic \(ax^2 + bx + c = 0\) is given by \(b^2 - 4ac\).

Here, \(a = c\), \(b = 3 - 2c\), and \(c = -(c + 3)\).

Calculate the discriminant: \((3 - 2c)^2 - 4c(-c - 3)\).

Simplify: \((3 - 2c)^2 + 4c(c + 3) = 9 - 12c + 4c^2 + 4c^2 + 12c = 8c^2 + 9\).

The discriminant \(8c^2 + 9\) is always positive for all values of \(c\), indicating two real roots.

Thus, the line and curve intersect for all values of \(c\).

To find the value of \(k\), we equate the curve and the line:

\((2k - 3)x^2 - kx - (k - 2) = 3x - 4\)

Rearrange to form a quadratic equation:

\((2k - 3)x^2 - (k + 3)x - (k - 6) = 0\)

For the line to be tangent to the curve, the discriminant of this quadratic must be zero:

\(b^2 - 4ac = 0\)

Here, \(a = 2k - 3\), \(b = -(k + 3)\), \(c = -(k - 6)\).

Calculate the discriminant:

\((k + 3)^2 + 4(2k - 3)(k - 6) = 0\)

Simplify:

\(k^2 + 6k + 9 + 8k^2 -60k + 72 = 0\)

\(9k^2 - 54k +81 = 0\)

Divide by 9:

\(k^2 - 6k + 9 = 0\)

\(k = 3\)

Since the line is tangent, \(k = 3\).

To find the points of intersection, set the equations equal: \(3x + k = x^2 + kx + 6\).

Rearrange to form a quadratic equation: \(x^2 + (k - 3)x + (6 - k) = 0\).

For two distinct points of intersection, the discriminant must be greater than zero: \(b^2 - 4ac > 0\).

Here, \(a = 1\), \(b = k - 3\), and \(c = 6 - k\).

Calculate the discriminant: \((k - 3)^2 - 4(6 - k) > 0\).

Simplify: \(k^2 - 2k - 15 > 0\).

Factorize: \((k + 3)(k - 5) > 0\).

The solution to this inequality is \(k < -3\) or \(k > 5\).

To find the points of intersection, set the equations equal: \(3x^2 - 4x + 4 = mx + m - 1\).

Rearrange to form a quadratic equation: \(3x^2 - (4 + m)x + (5 - m) = 0\).

For two distinct points of intersection, the discriminant \(b^2 - 4ac\) must be greater than zero.

Here, \(a = 3\), \(b = -(4 + m)\), and \(c = 5 - m\).

Calculate the discriminant: \((4 + m)^2 - 4 imes 3 imes (5 - m)\).

Simplify: \((4 + m)^2 - 12(5 - m) = m^2 + 20m - 44\).

Set the discriminant greater than zero: \(m^2 + 20m - 44 > 0\).

Factorize: \((m + 22)(m - 2) > 0\).

Using the sign chart method, the solution is \(m > 2\) and \(m < -22\).

To find the points of intersection, set the equations equal:

\(2x^2 + m(2x + 1) = 6x + 4\).

Rearrange to form a quadratic equation:

\(2x^2 + 2mx + m - 6x - 4 = 0\).

Simplify to:

\(2x^2 + (2m - 6)x + (m - 4) = 0\).

Use the discriminant \(b^2 - 4ac\) to determine the nature of the roots. Here, \(a = 2\), \(b = 2m - 6\), \(c = m - 4\).

The discriminant is:

\((2m - 6)^2 - 4(2)(m - 4)\).

Calculate:

\(4m^2 - 24m + 36 - 8m + 32\).

Simplify to:

\(4m^2 - 32m + 68\).

Factor or complete the square:

\((2m - 8)^2 + 1\).

This expression is always positive, indicating two distinct real roots for all values of \(m\).

Thus, the line intersects the curve at two distinct points for all values of \(m\).