Question

If $\alpha ,\beta$ be the roots of $x^2+x+2=0$ and $\gamma ,\delta$ be the roots of $x^2+3x+4=0$, then $(\alpha +\gamma )(\alpha +\delta )(\beta +\gamma )(\beta +\delta )$ is equal to

• A. $-18$
• B. $18$
• C. $24$
• D. $44$

Answer 1

The correct option is D $44$

$\alpha +\beta =-1,\alpha \beta =2,\gamma +\delta =-3,\gamma \delta =4$

Now, $(\alpha +\gamma )(\alpha +\delta )(\beta +\gamma )(\beta +\delta )$

$=({\alpha }^2+(-3)\alpha +4)({\beta }^2-3\beta +4)$

$=(-\alpha -2-3\alpha +4)(-\beta -2-3\beta +4)$

$=(-4\alpha +2)(-4\beta +2)$

$=4(2\alpha -1)(2\beta -1)$

$=4(4\alpha \beta -2\alpha -2\beta +1)$

$=4(8+2+1)=44$