The level of the polynomial in the numerator is 4 and also the level of the polynomial in the is 3. After that, the ratio will certainly be a polynomial of level 4-3= 1.

Therefore, the ratio is a straight expression and also its partial portions have the list below kind:

$$ frac {3x ^ 4 +7 x ^ 3 +8 x ^ 2 +53 x-186} {(x +4)( x ^ 2 +9)} =Ax+ B+ frac {C} {x +4} +frac {Dx+ E} {x ^ 2 +9} $$

When we increase the entire expression by $latex (x +4)( x ^ 2 +9)$, we have:

$$ 3x ^ 4 +7 x ^ 3 +8 x ^ 2 +53 x-186=( Ax+ B)( x +4)( x ^ 2 +9)+ C( x ^ 2 +9)+( Dx+ E)( x +4)$$

The worth of A is located by contrasting the coefficients of the terms with $latex x ^ 4$, and also we locate that $latex A= 3$.

Contrasting the coefficients of the $latex x ^ 3$ terms, we have:

$ latex 7= 4A+ B$

Substituting $latex A= 3$, we locate that $latex B= -5$.

When we make use of the worth $latex x= -4$, we have:

$$ 3( -4 )^ 4 +7( -4 )^ 3 +8( -4 )^ 2 +53( -4 )-186= C(( -4 )^ 2 +9)$$

$ latex 50= 25C$

$ latex C= 2$

Comparing the coefficients of the $latex x ^ 2$ terms, we have:

$$ 8= 9A +4 B+C+D$$

Substituting the worths $latex A= 4$, $latex B= -5$ and also $latex C= 2$, we have $latex D= -1$.

Ultimately, contrasting the continuous terms, we have:

$ latex -186= 36B +9 C +4 E$

Substituting the worths $latex B= -5$ and also $latex C= 2$, we have $latex E= -6$.

As a result, we have:

$$ frac {3x ^ 4 +7 x ^ 3 +8 x ^ 2 +53 x-186} {(x +4)( x ^ 2 +9)} =3x-5+ frac {2} {x +4} +frac {-x-6} {x ^ 2 +9} $$