Find the values of ‘m’ for which x² + 3xy + x + my –m has two linear factors in x and y, with integer coefficients.

For the given 2 degree equation

That must be equal to(ax + by + c)(dx + e)

= ad.x² + bd.xy + cd.x + ea.x + be.y + ec

= ad.x² + bd.xy + (cd + ea).x + be.y + ec

x² + 3xy + x + my–m = ad.x² + bd.xy + (cd + ea).x + be.y + ec

compare the equation

and take out the coefficient of every term

a.d = 1 ----------1

b.d = 3 ----------2

c.d + e.a = 1 ----------3

b.e = m ----------4

e.c = -m ----------5

⇒ from eq 1; a = d = 1 ∵ all coefficient are integers

After putting result in eq 3; c + e = 1 -------6

After putting result in eq 2; b = 3 --------7

⇒ divide eq 4 and 5

∴ that implies b = -c = -3 ∵ eq 7

Put value of c in eq 6

-3 + e = 1

e = 1 + 3 = 4

Putting value of b and e in eq 4

m = b × e

m = 3 × 4 = 12