For what values of $m$ will the expression $y^{2} + 2 x y + 2 x + m y - 3$ be capable of resolution into two rational factors?

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Solution

For a expression in the form $a x^{2} + b y^{2} + 2 h x y + 2 g x + 2 f y + c = 0$ to have two rational factors, then
$∣ ∣ ∣ ∣ \begin{matrix} a & h & g h & b & f g & f & c \end{matrix} ∣ ∣ ∣ ∣ = 0$
$\Rightarrow ∣ ∣ ∣ ∣ \begin{matrix} 0 & 1 & 1 1 & 1 & m 1 & m & - 3 \end{matrix} ∣ ∣ ∣ ∣ = 0$
$\Rightarrow - 1 (- 3 - m) + 1 (m - 1) = 0$
$\Rightarrow m + 3 + m - 1 = 0$
$\Rightarrow m = - 1$

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