Explain how to factor the following trinomials forms: x2 + bx + c and ax2 + bx + c. Is there more than one way to factor this? Show your answer using both words and mathematical notation

Salman Khan does his usual excellent work here

http://www.youtube.com/watch?v=nOZTe8jU2g4
for x^2 + bx +c , and
http://www.khanacademy.org/math/algebra/polynomials/v/factoring-trinomials-with-a-non-1-leading-coefficient-by-grouping
for ax^2 + bx + c

To factor trinomials of the form x^2 + bx + c and ax^2 + bx + c, we need to find two binomials that, when multiplied together, give us the original trinomial.

For the trinomial x^2 + bx + c, we can factor it by looking for two numbers, say p and q, whose sum is b and product is c. Once we find p and q, we can factor the trinomial as (x + p)(x + q).

For example, let's factor the trinomial x^2 + 7x + 10. We need to find two numbers whose sum is 7 and product is 10. Those numbers are 2 and 5, so we can factor the trinomial as (x + 2)(x + 5).

For the trinomial ax^2 + bx + c, we can apply the same method as before, but we also need to factor out the common factor a, if it exists. Then, we can factor the trinomial as a product of the common factor and two binomials.

For example, let's factor the trinomial 2x^2 + 6x + 4. First, we can see that there is a common factor of 2, so we factor it out: 2(x^2 + 3x + 2). Now we apply the method for x^2 + bx + c to the expression inside the parentheses. We need to find two numbers whose sum is 3 and product is 2. Those numbers are 1 and 2, so the factored form of the trinomial is 2(x + 1)(x + 2).

In summary, the steps to factor the trinomials x^2 + bx + c and ax^2 + bx + c are as follows:
1. For the trinomial ax^2 + bx + c, check if there is a common factor a and factor it out.
2. Apply the method for x^2 + bx + c to the remaining expression, if any, to find the two binomials.
3. Write the factored form as a product of the common factor and the two binomials.

Note that there may be cases where the trinomial cannot be factored using integers or simple factors. In such cases, you might need to use more advanced factoring techniques like factoring by grouping or quadratic formula.