factoring trinomials with a negative leading coefficient

Sure, here are the steps to factor a trinomial with a negative leading coefficient:

1. Write down the trinomial. For example, let's take -2x^2 + 7x - 3.

2. Factor out any common factors from all terms. In this case, there are no common factors, so we move to the next step.

3. Multiply the coefficient of x^2 (which is -2 in this case) by the constant term (which is -3 in this case). The result is 6.

4. Look for two numbers that multiply to give you 6 (the result from step 3) and add up to give you the coefficient of x (which is 7 in this case). The numbers that satisfy these conditions are 6 and 1.

5. Rewrite the middle term of the trinomial as the sum of the terms 6x and x. This gives us -2x^2 + 6x + x - 3.

6. Group the terms to make two binomials: (-2x^2 + 6x) + (x - 3).

7. Factor out the greatest common factor from each binomial. This gives us -2x(x - 3) + 1(x - 3).

8. Notice that (x - 3) is a common factor. Factor it out to get the final answer: (x - 3)(-2x + 1).

So, the factored form of -2x^2 + 7x - 3 is (x - 3)(-2x + 1).