construct a polynomial equation of lowest degree, with interger coefficients, and with roots, -1,-1, 2-3i

And this is what it looks like!

https://www.wolframalpha.com/input/?i=y%3D%28x+%2B+1%29%28x+%2B+1%29%28x+-%282-3i%29%29%28x+-+%282+%2B+3i%29%29

We have a problem here......we can't have a polynomial with only one complex root....all complex roots to a polymomial occur in pairs. Specifically, in your polynomial, the other complex root would be 2 + 3i.

I'll construct one with all these roots, and you can take your chances.   So we have....

(x + 1)(x + 1)(x -(2-3i))(x - (2 + 3i))

This gets a little messy, so hang on...multiplying the first two terms together, we get  x^2 + 2x + 1

Let's set this aside for the moment and concentrate on the last two .... so we have

(x -(2-3i))(x - (2 + 3i)) = (x^2 -x (2 - 3 i) - x (2 + 3i) + (2-3i)(2+3i) =

(x^2 -2x +3xi -2x -3xi + 4 -6i + 6i - 9(i)^2) = (x^2 - 4x + 4 + 9) = (x^2 - 4x + 13)

Now, putting the two together, we have

(x^2 + 2x + 1) (x^2 - 4x +13) =

x^4 + 2x^3 + x^2 - 4x^3 - 8x^2 - 4x + 13x^2 +26x + 13 =

x^4 -2x^3 + 6x^2 + 22x + 13 = P(x)

See...I told you it could get messy....!!

BTW...I checked this one with a solver and it was correct.......