The problem asks for the direction ratios of a line M that is parallel to a given plane P.

Plane P Analysis

The equation of the plane P is given as: $2x+3y+z+5=0\$\; The\; normal\; vector$ \overrightarrow{n}\$\; to\; the\; plane\; P\; is\; determined\; by\; the\; coefficients\; of\; x,\; y,\; and\; z.$$

Normal vector

Line M Condition

A line M is parallel to the plane P if its direction vector $\overrightarrow{m}\$\; is\; perpendicular\; to\; the\; plane\text{'}s\; normal\; vector$ \overrightarrow{n}.$$

The condition for perpendicularity is that their dot product is zero:

$\overrightarrow{m}·\overrightarrow{n}=0\$$

Let the direction ratios of line M be $<l,m,n>\$.\; Then,\; the\; condition\; becomes:$

$<l,m,n>·<2,3,1>=0\$$

Or, $2l+3m+n=0\$$

Checking Options

We test each option's direction ratios $<l,m,n>\$\; using\; the\; condition$ 2l+3m+n=0\$:$$

• Option 1: $<-3,2,1>\$$
  $2(-3)+3\left(2\right)+1=-6+6+1=1\ne 0\$$
• Option 2: $<3,2,-6>\$$
  $2\left(3\right)+3\left(2\right)+(-6)=6+6-6=6\ne 0\$$
• Option 3: $<1,3,2>\$$
  $2\left(1\right)+3\left(3\right)+2=2+9+2=13\ne 0\$$
• Option 4: $<2,2,-10>\$$
  $2\left(2\right)+3\left(2\right)+(-10)=4+6-10=10-10=0\$$

Option 4 satisfies the condition $\overrightarrow{m}·\overrightarrow{n}=\; 0$

Conclusion

The direction ratios $<2,2,-10>\$\; correspond\; to\; a\; line\; M\; parallel\; to\; the\; plane\; P.$