Answer:
[tex]g^2-4g-21=(g-7)(g+3)[/tex]
Step-by-step explanation:
To complete the left side of the equation, we need to bring it to the form
[tex](g-a)(g+b)[/tex]
expanding this expression we get:
[tex]g^2+bg-ag-ab[/tex]
[tex]g^2+(b-a)g-ab[/tex]
Thus we have
[tex]g^2-4g-21=g^2+(b-a)g-ab[/tex]
from here we see that for both sides of the equation to be equal, it must be that
[tex]b-a=-4[/tex]
[tex]-ab=-21[/tex].
Getting rid of the negative signs we get:
[tex]a-b=4[/tex]
[tex]ab=21[/tex]
At this point we can either guess the solution to this system (that's how you usually solve these types of problems) or solve for [tex]a[/tex] and [tex]b[/tex] systematically.
The solutions to this set are [tex]a=7[/tex] and [tex]b=3[/tex]. (you have to guess on this—it's easier)
Therefore, we have
[tex](g-a)(g+b)=(g-7)(g+3)[/tex]
which completes our equation
[tex]\boxed{ g^2-4g-21=(g-7)(g+3)}[/tex]
