manipulating two numbers that both have + and - sign in front

Question

I have encountered a math formula while studying adding or substracting variance of two independent distribution

My question is, why

and

equals

Both number have a + and - sign in front, and how they are manipulated is really tripping me up. Could someone please kindly explain?

Answer 1

$$\pm 2E(XY) \mp 2\mu_x\mu_y = \pm (2E(XY) - \mu_x\mu_y)$$

When the coefficient of $E(XY)$ is $+2$, the corresponding coefficient for $\mu_x\mu_y$ is $-2$.

$$+ 2E(XY) - 2\mu_x\mu_y = +(2E(XY) - \mu_x\mu_y)$$

When the coefficient of $E(XY)$ is $-2$, the corresponding coefficient for $\mu_x\mu_y$ is $+2$.

$$- 2E(XY) + 2\mu_x\mu_y = -(2E(XY) - \mu_x\mu_y)$$

Answer 2

First case: using the "top" signs. Then we have $\color{blue}{+2E(xy)}$ and $\color{blue}{-2\mu_x\mu_y}$, which add up to: $$+2E(xy)-2\mu_x\mu_y=+2[E(xy)-\mu_x\mu_y].$$ Second case: using the "bottom" signs. Then we have $\color{blue}{-2E(xy)}$ and $\color{blue}{+2\mu_x\mu_y}$, which add up to: $$-2E(xy)+2\mu_x\mu_y=-2[E(xy)-\mu_x\mu_y].$$ Putting the two cases together, we can abbreviate the outcome as $\pm2[E(xy)-\mu_x\mu_y]$, as claimed.