Option C:
[tex]2 g^{3}-5 g^{2}+6[/tex]
Solution:
Given expression:
[tex]$\frac{\left(5 g^{4}+5 g^{3}-17 g^{2}+6 g\right)-\left(3 g^{4}+6 g^{3}-7 g^{2}-12\right)}{g+2}[/tex]
To find which expression is equal to the given expression.
[tex]$\frac{\left(5 g^{4}+5 g^{3}-17 g^{2}+6 g\right)-\left(3 g^{4}+6 g^{3}-7 g^{2}-12\right)}{g+2}[/tex]
Expand the term [tex]-\left(3 g^{4}+6 g^{3}-7 g^{2}-12\right):-3 g^{4}-6 g^{3}+7 g^{2}+12[/tex]
[tex]$=\frac{5 g^{4}+5 g^{3}-17 g^{2}+6 g- 3 g^{4}-6 g^{3}+7 g^{2}+12}{g+2}[/tex]
Arrange the like terms together.
[tex]$=\frac{5 g^{4}- 3 g^{4}+5 g^{3}-6 g^{3}-17 g^{2}+7 g^{2}+6 g+12}{g+2}[/tex]
[tex]$=\frac{2 g^{4}- g^{3}-10 g^{2}+6 g+12}{g+2}[/tex]
Factor the numerator [tex]2 g^{4}-g^{3}-10 g^{2}+6 g+12=(g+2)\left(2 g^{3}-5 g^{2}+6\right)[/tex]
[tex]$=\frac{(g+2)\left(2 g^{3}-5 g^{2}+6\right)}{g+2}[/tex]
Cancel the common factor g + 2, we get
[tex]=2 g^{3}-5 g^{2}+6[/tex]
Hence option C is the correct answer.
