How do you factor by grouping #28s^2 - 37s - 21#?

Question

Kennedy Adams · Accepted Answer

Factoring #f(x) = 28x^2 - 37x - 21#. There are 2 methods. 1. Factoring AC method (factoring by grouping) . 
Find 2 numbers b1 and b2 that satisfy these 2 conditions:
Sum: #(b1 + b2) = -37#
and 
Product: # (b1*b2) = a*c = 588# To find #b1 " and "b2" "# compose factor pairs of  #a*c = -588#
Proceed: #(1, -588),(2,-294),....(12, -49)..."etc."# 
We find 
#b1 = 12" and "b2 = -49#
( since their sum is #-37#. ) Next factor by grouping:
#f(x) = 28x^2 - 49x + 12x - 21#
# = 7x*(4x - 7) + 3*(4x - 7)# Factored form:
f(x) = (4x - 7)(7x + 3)  2. The new AC method to factor a trinomial f(x) 
#f(x) = 28x^2 - 37x - 21."    (1)"# First convert trinomial (1) to 
trinomial: #f(x) = x^2 - 37x - 588 "          (2)#, 
with #a*c = -21*(28) = -588# Compose factor pairs of #a*c = -#
and apply the Rule of Sign for Real Roots.  Proceed: #(-1, 588)....(-12, 49)# This sum is #49 - 12 = 37 = -b# Then #b'1 = 12" and #b'2 = -49# Next, divide #b'1" and "b'2# by a 
to get #b1" and "b2# for trinomial (1). #b1 = b'1/a = 12/28 = 3/7 ,#
 and 
#b2 = b'2/a = -49/28 = -7/4.# Then, the factored form is: 
#f(x) = (x + 3/7)(x - 7/4)#
# = (7x + 3)(4x - 7)# This new AC Method avoids the lengthy factoring by grouping.

Genesis Allen · Answer

undefined To factor by grouping the expression $28s^2 - 37s - 21$, follow these steps:

1. Multiply the coefficient of the $s^2$ term (28) by the constant term (-21) to get -588.
2. Find two numbers that multiply to -588 and add up to the coefficient of the $s$ term (-37). These numbers are -49 and 12.
3. Rewrite the middle term (-37s) using the two numbers found in step 2: $28s^2 - 49s + 12s - 21$.
4. Group the terms: $(28s^2 - 49s) + (12s - 21)$.
5. Factor out the greatest common factor from each group: $7s(4s - 7) + 3(4s - 7)$.
6. Notice that both groups have a common factor of $(4s - 7)$.
7. Factor out the common factor: $(4s - 7)(7s + 3)$.

So, the factored form of $28s^2 - 37s - 21$ is $(4s - 7)(7s + 3)$.