Answer:
A, C, E
Step-by-step explanation:
By analyzing the graph, we can see that the roots are [tex]-2[/tex] and [tex]-4[/tex].
We now want to find all equations of the form [tex](x-(-2))(x-(-4))=(x+2)(x+4)=0[/tex].
This immediately gives C as 1 answer.
We can expand our equation to find other possible solutions!
By the FOIL method: [tex](x+2)(x+4)=(x)(x)+(2)(x)+(4)(x)+(2)(4)=x^2+6x+8=0[/tex]
We can then divide both sides of the equation to find that [tex]-x^2-6x-8=0[/tex], so A is another answer.
We can complete the square on [tex]x^2+6x+8[/tex] to find the last possible solution.
[tex]x^2+6x+8=x^2+6x+9-9+8=x^2+6x+9-1=(x+3)^2-1=0[/tex].
We can then multiply both sides of the equation by [tex]4[/tex] to find that [tex]4(x+3)^2-4=0[/tex], so E is the last answer.
The proof that the other answers don't work is left as an exercise.
So, the answers are A, C, and E, and we're done!
