If {x^2} + 2x + 2xy + my - 3 has two rational | vedclass

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(C) The given expression is $f(x, y) = x^2 + 2x(1 + y) + (my - 3)$.For the expression to have rational factors,its discriminant $D$ with respect to $x

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$6, - 2$

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(C) The given expression is $f(x, y) = x^2 + 2x(1 + y) + (my - 3)$.For the expression to have rational factors,its discriminant $D$ with respect to $x$ must be a perfect square.The discriminant $D$ is given by $D = [2(1 + y)]^2 - 4(1)(my - 3)$.$D = 4(1 + y)^2 - 4(my - 3) = 4(1 + y^2 + 2y - my + 3) = 4(y^2 + (2 - m)y + 4)$.For $D$ to be a perfect square,the quadratic expression $y^2 + (2 - m)y + 4$ must be a perfect square of the form $(y \pm 2)^2$.Expanding $(y \pm 2)^2$,we get $y^2 \pm 4y + 4$.Comparing $y^2 + (2 - m)y + 4$ with $y^2 \pm 4y + 4$,we have $2 - m = \pm 4$.Case $1$: $2 - m = 4 \Rightarrow m = -2$.Case $2$: $2 - m = -4 \Rightarrow m = 6$.Thus,the values of $m$ are $6$ and $-2$.

If ${x^2} + 2x + 2xy + my - 3$ has two rational factors,then the value of $m$ will be

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