Find a polynomial, which, when added to the polynomial 5x2–3x–9, is equivalent to: 2x−3

Step-by-step explanation:

The best way to do this is to subtract the 2x-3 from the polynomial to find the difference.  The difference is what needs to be added to the polynomial to get 2x-3 as the sum.

5x² - 3x - 9 - (2x - 3) =

5x² - 3x - 9 - 2x + 3 =

5x² - 5x - 6

This should be what we need to add to the polynomial given to give us a sum of 2x - 3.  But if we add 5x² to another 5x² we get 10x².  We want to eliminate the x²-ed terms altogether, so let's take the opposite of our sum and see what happens when we add.  The opposite of our sum is

-5x² + 5x + 6.  Now let's add:

-5x² + 5x + 6 + 5x² - 3x - 9 =

2x - 3.  Therefore, the polynomial to add in is

-5x² + 5x + 6