Step-by-Step Solution

1. Write the equations of the lines in parametric form:<\/strong><\/p>

   Line 1: \(\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-1}{5} = \lambda\)
   \(x = 3\lambda + 1\), \(y = 2\lambda - 1\), \(z = 5\lambda + 1\)

   Line 2: \(\frac{x+2}{4}=\frac{y-1}{3}=\frac{z+1}{-2} = \mu\)
   \(x = 4\mu - 2\), \(y = 3\mu + 1\), \(z = -2\mu - 1\)

2. Equate the coordinates to find the point of intersection (if it exists):<\/strong><\/p>

   \(3\lambda + 1 = 4\mu - 2\) (1)
   \(2\lambda - 1 = 3\mu + 1\) (2)
   \(5\lambda + 1 = -2\mu - 1\) (3)

3. Solve equations (1) and (2) for \(\lambda\) and \(\mu\):<\/strong><\/p>

   From (1): \(3\lambda - 4\mu = -3\) (4)
   From (2): \(2\lambda - 3\mu = 2\) (5)

   Multiply (4) by 2 and (5) by 3:
   \(6\lambda - 8\mu = -6\) (6)
   \(6\lambda - 9\mu = 6\) (7)

   Subtract (6) from (7):
   \(-\mu = 12\)
   \(\mu = -12\)

   Substitute \(\mu = -12\) into (5):
   \(2\lambda - 3(-12) = 2\)
   \(2\lambda + 36 = 2\)
   \(2\lambda = -34\)
   \(\lambda = -17\)

4. Check if these values of \(\lambda\) and \(\mu\) satisfy equation (3):<\/strong><\/p>

   Substitute \(\lambda = -17\) and \(\mu = -12\) into (3):
   \(5(-17) + 1 = -2(-12) - 1\)
   \(-85 + 1 = 24 - 1\)
   \(-84 = 23\)

   This is not true, so the lines do not intersect.