Question

Using complete sentences, explain how to completely factor 2x2 - 8x - 42.

Answer 1

first factor out a 2, then.......

Answer 2

Consider a general quadratic equation written as follows:

ax^2 +bx + c
where a, b and c are constants.

one quick way of factorizing this is to split the bx term using the following criteria:

find two numbers d and h such that
d+h = b and
d*h = a*c

=> ax^2 + bx +c = ax^2 + (d+h)x + c (i.e replace b with (d+h))

ax^2 + bx +c = ax^2 + (d+h)x + c= ax^2 + dx + hx + c
factorize the above equation and group
the factors.

For your example:
If you first factor 2 as suggested by lag: then rewrite the equation as follows:

2x^2-8x - 42 = 2*(x^2 -4x -21)
Factorize the equation in the brackets by comparing with ax^2 + bx +c:

x^2 -4x -21 => b = -4, a = 1, c = -21
choose d= -7 and h =3
i.e. d+h = -7+3 = -4 = b
and d*h = -7*3 = -21 = C

so re-write
x^2 -7x -21 = x^2 -7x + 3x -21
factorize
x^2 -7x + 3x -21 = x(x-7) + 3(x-7)
group: the factors

(x+3)(x-7)

hence x^2-7x-21 = (x+3)(x-7)
since you had factored 2 originally
the final answer is

2*(x+3)(x-7)