If $f(x)=2 x^4-13 x^2+a x+b$ is divisible by $x^2-3 x+2$,then $(a, b)$ is equal to

The sum of the non-real roots of $(p^2+p-3)(p^2+p-2)-12=0$ is

Let $R^2$ denote $R \times R$. Let $S = \{(a, b, c) : a, b, c \in R \text{ and } ax^2 + 2bxy + cy^2 > 0 \text{ for all } (x, y) \in R^2 - \{(0, 0)\}\}$. Then which of the following statements is (are) $TRUE$?
$(A) (2, \frac{7}{2}, 6) \in S$
$(B) \text{If } (3, b, \frac{1}{12}) \in S, \text{ then } |2b| < 1$
$(C) \text{For any given } (a, b, c) \in S, \text{ the system of linear equations } ax + by = 1, bx + cy = -1 \text{ has a unique solution.}$
$(D) \text{For any given } (a, b, c) \in S, \text{ the system of linear equations } (a+1)x + by = 0, bx + (c+1)y = 0 \text{ has a unique solution.}$

Let $f(x) = Ax^2 + Bx$ and $g(x) = Lx^2 + Mx + N$. Given that $f(2) - g(2) = 1$, $f(3) - g(3) = 4$, and $f(4) - g(4) = 9$. Then a root of $f(x) - g(x) = 0$ is

Solve the equation $x^{2}+\frac{x}{\sqrt{2}}+1=0$.

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