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Mathematics
Problem
Consider the rational function
5
x
2
−
3
(
x
+
1
)
2
(
x
2
+
4
)
\dfrac{5x^{2}-3}{(x+1)^{2}(x^{2}+4)}
(
x
+
1
)
2
(
x
2
+
4
)
5
x
2
−
3
. Which of the following decompositions is structurally correct? a)
A
x
+
1
+
B
(
x
+
1
)
2
+
C
x
2
+
4
\dfrac{A}{x+1}+\dfrac{B}{(x+1)^{2}}+\dfrac{C}{x^{2}+4}
x
+
1
A
+
(
x
+
1
)
2
B
+
x
2
+
4
C
b)
A
x
+
1
+
B
(
x
+
1
)
2
+
C
x
+
D
x
2
+
4
\dfrac{A}{x+1}+\dfrac{B}{(x+1)^{2}}+\dfrac{Cx+D}{x^{2}+4}
x
+
1
A
+
(
x
+
1
)
2
B
+
x
2
+
4
C
x
+
D
c)
A
x
+
1
+
B
(
x
+
1
)
2
+
C
x
+
D
(
x
2
+
4
)
2
\dfrac{A}{x+1}+\dfrac{B}{(x+1)^{2}}+\dfrac{Cx+D}{(x^{2}+4)^{2}}
x
+
1
A
+
(
x
+
1
)
2
B
+
(
x
2
+
4
)
2
C
x
+
D
d)
A
x
+
1
+
C
x
+
D
x
2
+
4
\dfrac{A}{x+1}+\dfrac{Cx+D}{x^{2}+4}
x
+
1
A
+
x
2
+
4
C
x
+
D
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