What adds -1y but multiply by -42y2?

Answer 1

I'm assuming you mean what two expressions sum to -1y
and also whose produce is -42y2?(when you can't use formatting to make your 2 higher, you can use the carrot symbol to signify you are raising something to the power: i.e., -42y^2)

We can save a little time and realize:

ay + by = (a+b)y <-- the sum yields just a single y term (matches problem)

(ay)(by) = (ab)(y2) <-- the product yields a y squared term (matches problem)

Because of that, we can drop the y's.

We can then say:
a + b = -1

ab = -42

We have two equations, and two variables, that's a good thing.

If we solve the first equation for a we get:
a+b= -1
a = -1-b

Plug it into the second equation:
ab = -42
(-1-b)b = -42
-b+b2 = -42

b2- b + 42= 0 (we now have a polynomial)

(Once you get comfortable with this, you can jump straight to this step from the problem)

You need to figure out what two numbers have a product of42 and whose sum is -1. Alternatively you could use the quadratic equation (but most people try everything to avoid using it).

(1)(-42) = -42; 1-42 = -41
(2)(-21) = -42; 2-21 = -19
(6)(-7) = -42; 6-7 = -1 (bingo, we got something that works)

Remember how we dropped the y's on the numbers though, we need to readd them at this point.

6y and -7y

Let's check our work, to see if they add and multiply correctly:

(6y) + (-7y) = 6y - 7y = -1y (check)

(6y)(-7y) = -42y2 (check)

So your answer is:
6y and -7y