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4x^2 + 4y^2 - 4x + 12y - 6  = 0     add 6 to both sides

4x^2  - 4x + 4y^2 + 12y   =  6        complete the square on x and y

4(x^2 - x + 1/4)  + 4 (y^2 + 3y + 9/4)   =  6  + 1  + 9

4(x - 1/2)^2  +  4(y + 3/2)^2  =  16       divide through by 4

(x - 1/2)^2  + ( y + 3/2)^2  =   4   ⇒   "D"

C  will

(6 - 4i)  - (2 + 4i)  =

6 - 2  - 4i - 4i  =

4  - 8i

Nvm, I got 40 thanks everyone ^-^

Your answer is correct but I think there's more than one possible answer. 

I used the program called Desmos to help me with this task and I saw that there is more than one correct answer.

Image of graphs -->https://www.desmos.com/calculator/rpf7nusezm

Thanks.

|x+y-4|=5

A point is selected at random inside the given figure. What is the probability the point will be in the region labeled D? Choose your answer, as a fraction in simplest form. P(D) = 

That's how I got 42 but idk about 40

Is it https://www.desmos.com/calculator/tfl8wyadtm

Ty I really appreciate your help!

Welcome! Solving them right now! 

Ty so so much tertre and Omi! Now I need help understanding 40 and 42 lol

Same thing as we did in 41.

Add 2 to both sides, to get, \(|2x+1|\geq7\)

|f(x)| greater than or equal to a, f(x) less than or equal to a, or f(x) greater than or equal to a

\(2x+1\leq-7: x\leq-4\)

\(2x+1\geq7: x\geq3\)

Combine the ranges, to get \(x\leq-4 \) or \(x\geq3\)

Number 41.

41. First add 8 to both sides, to get,  |3x+6|=6

Then, we have, |f(x)|=a - f(x)=-a or f(x)=a

So, 3x+6=-6 or 3x+6=6

3x+6=-6: x=-4

3x+6=6: x=0

So, x=-4, or x=0

Please help asap!