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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} \$\$