## 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 add or 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 $$x^2 + 7x + 12$$.

1. Factor the first term: $$x^2 = x \cdot x$$.
2. Factor the last term: $$12 = 3 \cdot 4$$ and $$3 + 4 = 7$$, which is our middle term.
3. Determine the signs: Both signs are positive because $$+3 + 4 = +7$$.
4. Check by multiplying: $$(x + 3)(x + 4) = x^2 + 7x + 12$$.

So, $$x^2 + 7x + 12 = (x + 3)(x + 4)$$.

### Exercises:

Factor the following trinomials:

1. $$x^2 + 5x + 6$$
2. $$x^2 + 9x + 20$$
3. $$x^2 + 7x + 10$$
4. $$x^2 + 11x + 30$$
5. $$x^2 + 8x + 15$$
6. $$x^2 + 10x + 21$$
7. $$x^2 + 6x + 8$$
8. $$x^2 + 12x + 35$$
9. $$x^2 + 15x + 56$$
10. $$x^2 + 13x + 40$$

### Solutions:

1. $$x^2 + 5x + 6 = (x + 3)(x + 2)$$
2. $$x^2 + 9x + 20 = (x + 5)(x + 4)$$
3. $$x^2 + 7x + 10 = (x + 5)(x + 2)$$
4. $$x^2 + 11x + 30 = (x + 6)(x + 5)$$
5. $$x^2 + 8x + 15 = (x + 5)(x + 3)$$
6. $$x^2 + 10x + 21 = (x + 7)(x + 3)$$
7. $$x^2 + 6x + 8 = (x + 4)(x + 2)$$
8. $$x^2 + 12x + 35 = (x + 7)(x + 5)$$
9. $$x^2 + 15x + 56 = (x + 8)(x + 7)$$
10. $$x^2 + 13x + 40 = (x + 8)(x + 5)$$

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