## Factoring Trinomials of the Form $$x^2 - bx - c$$

Factoring trinomials, which are polynomials with three terms, can initially seem challenging. However, with practice, it becomes routine. If a trinomial factors, it will factor into the product of two binomials.

### Steps to Factor Trinomials:

1. Factor the first term: $$x^2 = x \cdot x$$.
2. Factor the last term: Choose factors that multiply to obtain the last term and subtract to obtain the middle term.
3. Determine the signs: by adding or subtracting the product of the inner and outer terms.
4. Check by multiplying.

### Example:

Factor the trinomial $$x^2 - 3x - 10$$.

1. Factor the first term: $$x^2 = x \cdot x$$.
2. Factor the last term: $$-10 = -5 \cdot 2$$ and $$-5 + 2 = -3$$, which is our middle term.
3. Determine the signs: One sign is positive and the other is negative because $$-5 + 2 = -3$$.
4. Check by multiplying: $$(x - 5)(x + 2) = x^2 - 3x - 10$$.

So, $$x^2 - 3x - 10 = (x - 5)(x + 2)$$.

### Exercises:

Factor the following trinomials:

1. $$x^2 - 5x - 6$$
2. $$x^2 - 7x + 10$$
3. $$x^2 - 6x - 16$$
4. $$x^2 - 8x - 9$$
5. $$x^2 - 3x - 10$$
6. $$x^2 - 4x - 21$$
7. $$x^2 - 5x - 14$$
8. $$x^2 - 6x - 16$$
9. $$x^2 - 7x - 18$$
10. $$x^2 - 9x - 22$$

### Solutions:

1. $$x^2 - 5x - 6 = (x - 6)(x + 1)$$
2. $$x^2 - 7x + 10 = (x - 2)(x - 5)$$
3. $$x^2 - 6x - 16 = (x - 8)(x + 2)$$
4. $$x^2 - 8x - 9 = (x - 9)(x + 1)$$
5. $$x^2 - 3x - 10 = (x - 5)(x + 2)$$
6. $$x^2 - 4x - 21 = (x - 7)(x + 3)$$
7. $$x^2 - 5x - 14 = (x - 7)(x + 2)$$
8. $$x^2 - 6x - 16 = (x - 8)(x + 2)$$
9. $$x^2 - 7x - 18 = (x - 9)(x + 2)$$
10. $$x^2 - 9x - 22 = (x - 11)(x + 2)$$

Remember to always check your results by multiplying the factors. If the product matches the original trinomial, then the factoring is correct.