We can factor quadratic equations of the form $ax^2 + bx + c$ by first finding the factors of the constant $c$.  The factored form of a quadratic equation takes the general form:

$(\alpha_1 x + \beta_1)(\alpha_2 x + \beta_2)$

When $a$ is equal to one, $\alpha_1$ and $\alpha_2$ both equal one, and $\beta_1$ and $\beta_2$ are factors of the constant $c$ such that:

$b = \beta_1 + \beta_2$

When $a$ is not equal to one and not equal to zero, you can FOIL the above expression for the factored form of the quadratic to find that  $\alpha_1$ and $\alpha_2$ are factors of $a$ such that:

$a=\alpha_1 \alpha_2$ and $b = \alpha_1 \beta_2 + \alpha_2 \beta_1$

In other words, the coefficient of the $x^2$ term is given by the product of the coefficients $\alpha_1$ and $\alpha_2$, and the coefficient of the $x$ term is given by the inner and outer parts of the FOIL process.

In some cases, it will be impossible to factor the quadratic such that $\alpha_1$ and $\alpha_2$ are integers. 

Example: $a=1$

Let the quadratic equation be:

$-4x + x^2 - 21$

To factor this, we have to arrange the quadratic equation in order of largest exponent value to smallest exponent value.  

$x^2-4x-21$

Next, we identify the constants $b$ and $c$, which in this case are $-4$ and $-21$, respectively.  

Then we list the possible pairs of factors of the constant $c$, which yields the sets 1 and -21, -1 and 21, 3 and -7, and -3 and 7. Next we need to find the factored set of values that add to equal the value of $b$.  In this case, the correct values are 3 and -7, since they add to equal $-4$.  This leads to the factored form:

$x^2 - 4x - 21 = (x - 7)(x + 3)$

Example: $a \neq 1$

Let the quadratic equation be:

$6x^2+16x+8$

First, we factor $a$, which has one pair of factors 3 and 2. Then we factor the constant $c$, which has one pair of factors 2 and 4. Using these factored sets, we assemble the final factored form of the quadratic 

$(\alpha_1 x + \beta_1)(\alpha_2 x + \beta_2)$ 

Such that $b = \alpha_1 \beta_2 + \alpha_2 \beta_1$.  This leads to the equation:

$(3x +2)(2x+4)$