When we have a polynomial of the form
#ax^2+bx+c#
We can factor this quadratic with splitting up the #b# term into two terms. This allows us to factor the left side of the expression and the right side individually, and look for a common factor between them. This is factoring by grouping.
Let's take our polynomial
#ax^2+bx+c# again. To factor by grouping, we can rewrite this expression as
#color(blue)(a)x^2+bx+color(red)(c)=color(blue)ax^2+(color(blue)a+color(red)c)x+color(red)c#
Notice that #(a+c)x# is the same as our #b# term. We can distribute the #x# to both terms to get
#color(blue)(ax^2+ax)+color(red)(cx+c)#
This is the essence of factoring by grouping. We can look at our polynomial as two groups of two terms.
From the blue terms, we can factor out an #ax#, and from the red terms, a #c#. This leaves us with
#color(blue)(ax)color(purple)((x+1))+color(red)c color(purple)((x+1))#
Now, both terms have an #x+1# in common, so we can factor that out to get
#(color(blue)(ax)+color(red)(c))color(purple)((x+1))#
We will not always have an #x+1# term. For instance, take the following polynomial:
#3x^2+11x+6#
Let's rewrite this as
#color(turquoise)(3x^2+9x)+color(orange)(2x+6)#
We can factor a #3x# out of the blue terms, and a #2# out of the orange terms. We get
#color(turquoise)(3x)(x+3)+color(orange)2(x+3)#
We can factor an #x+3# out to get
#(3x+2)(x+3)#
The key point is that we can rewrite our #b# term as the sum of two terms so we can factor twice. Next, we look for a common factor between our newly factored expression.
Factoring by grouping will not always work- at this point it may be a good idea to resort to the Quadratic Formula.
Hope this helps!
