Answer:
[tex]\frac{6x^2+12x-177}{x^2+x -30}=6+\frac{3}{x-5}+\frac{3}{x+6}[/tex]
Step-by-step explanation:
From the question we are told that:
Partial fraction is given as
[tex]\frac{(6x^2+12x-177)}{(x^2+x-30)}[/tex]
Factorized
[tex]6+\frac{6x+3}{x^2+x−30}[/tex]
[tex]\frac{6x+3}{(x-5)(x+6)}[/tex]
Generally the equation for Partial Fraction is mathematically given by
[tex]\frac{6x+3}{(x-5)(x+6)}=\frac{A}{x-5}+\frac{B}{x+6}[/tex]
Therefore
[tex]\frac{6x+3}{(x-5)(x+6)}=\frac{(x-5)B+(x+6)A}{(x-5)(x+6)}[/tex]
Since denominators are equal
[tex]6x+3=(x-5)B+(x+6)A[/tex]
[tex]6x+3=xA+xB+6A-5B[/tex]
[tex]6x+3=x(A+B)+6A-5B[/tex]
Collecting Coefficients respectively
[tex]A+B=6 .......(equ 1)[/tex]
[tex]6A- 5B=3.........(equ 2)[/tex]
Therefore
A=3
B=3
Hence, Partial fraction decomposition is
[tex]\frac{6x^2+12x-177}{x^2+x -30}=6+\frac{3}{x-5}+\frac{3}{x+6}[/tex]
