Please help factor this polynomial?

Question

Can anyone help factor this polynomial completely?

x^2 + xy +3x - 2y^2 + 2y +2.

The closest I've got is:

(x + 2y + 1)(x - y +2) which gives me x^2 +xy + 3x - 2y^2 + 3y + 2.

The only difference in my answer is that the second last term should be 2y not 3y.

If anyone could help that would be wonderful!!!

Answer 1

Suppose there is a factor (x - ay - b). Then if we substitute x = ay + b, the polynomial must be identically 0 for all y.
(ay+b)^2 + (ay+b)y + 3(ay+b) - 2y^2 + 2y + 2 = 0
<=> a^2y^2 + 2aby + b^2 + ay^2 + by + 3ay + 3b - 2y^2 + 2y + 2 = 0
<=> y^2(a^2 + a - 2) + y(2ab + b + 3a + 2) + (b^2 + 3b + 2) = 0
Since this must be 0 for all y, we must have all the coefficients 0:
<=> a^2 + a - 2 = 0, 2ab + b + 3a + 2 = 0, b^2 + 3b + 2 = 0
The first equation gives us (a+2)(a-1) = 0 and hence a = 1 or -2.
The last equation gives us (b+1)(b+2) = 0 and hence b = -1 or -2.
a = 1, b = -1 gives 2ab + b + 3a + 2 = 2
a = 1, b = -2 gives 2ab + b + 3a + 2 = -1
a = -2, b = -1 gives 2ab + b + 3a + 2 = -1
a = -2, b = -2 gives 2ab + b + 3a + 2 = 2
Alternatively, substituting a = 1 into 2ab + b + 3a + 2 = 0 gives b = -5/3 and substituting a = -2 gives b = -4/3. In either case, there are no values of a and b which satisfy all three equations. So there are no factors of form (x - ay - b).

In other words, this polynomial isn't factorisable into linear factors.

Answer 2

The polynomial x^2 + xy +3x - 2y^2 + 2y +2 cannot be further factorised.

Answer 3

there is no answer to it.pls check whether it is the correct question.

Answer 4

No answer to this question!