`(x^(3)+3x^(2)+5)/((x-1)(x+2)^2) =1+(a)/(x-1)+(b)/(x+2)+(c)/((x+2)^2)`
a, b, c are respectively

Options

A. 1,2,3
B. 1,-1,3
C. 1,-1,-3
D. 1,0,3

Answer

To solve the equation

\[ \frac{x^3 + 3x^2 + 5}{(x-1)(x+2)^2} = 1 + \frac{a}{x-1} + \frac{b}{x+2} + \frac{c}{(x+2)^2} \]

we will find the values of \(a\), \(b\), and \(c\).

### Step 1: Rewrite the Right-Hand Side with a Common Denominator

The common denominator for the right-hand side is \((x-1)(x+2)^2\). Thus, we rewrite the right-hand side:

\[ 1 + \frac{a}{x-1} + \frac{b}{x+2} + \frac{c}{(x+2)^2} = \frac{(x-1)(x+2)^2 + a(x+2)^2 + b(x-1)(x+2) + c(x-1)}{(x-1)(x+2)^2} \]

### Step 2: Expand the Numerator

Now, we need to expand the numerator:

1. Expand \((x-1)(x+2)^2\):
\[ (x-1)(x^2 + 4x + 4) = x^3 + 4x^2 + 4x - x^2 - 4x - 4 = x^3 + 3x^2 - 4 \]

2. Expand \(a(x+2)^2\):
\[ a(x^2 + 4x + 4) = ax^2 + 4ax + 4a \]

3. Expand \(b(x-1)(x+2)\):
\[ b(x^2 + x - 2) = bx^2 + bx - 2b \]

4. Expand \(c(x-1)\):
\[ cx - c \]

Combining all these, the numerator becomes:
\[ x^3 + 3x^2 - 4 + ax^2 + 4ax + 4a + bx^2 + bx - 2b + cx - c \]

### Step 3: Combine Like Terms

Combining like terms gives us:
\[ x^3 + (3 + a + b)x^2 + (4a + b + c)x + (4a - 2b - c) \]

### Step 4: Set the Numerators Equal

Now, we set the numerator of the left-hand side equal to the numerator of the right-hand side:
\[ x^3 + 3x^2 + 5 = x^3 + (3 + a + b)x^2 + (4a + b + c)x + (4a - 2b - c) \]

### Step 5: Equate Coefficients

From the above equation, we can equate the coefficients:

1. Coefficient of \(x^2\):
\[ 3 + a + b = 3 \implies a + b = 0 \quad (1) \]

2. Coefficient of \(x\):
\[ 4a + b + c = 0 \quad (2) \]

3. Constant term:
\[ 4a - 2b - c = 5 \quad (3) \]

### Step 6: Solve the System of Equations

From equation (1):
\[ b = -a \quad (4) \]

Substituting (4) into (2):
\[ 4a - a + c = 0 \implies 3a + c = 0 \implies c = -3a \quad (5) \]

Substituting (4) and (5) into (3):
\[ 4a - 2(-a) - (-3a) = 5 \implies 4a + 2a + 3a = 5 \implies 9a = 5 \implies a = \frac{5}{9} \]

Now substituting \(a\) back into (4) to find \(b\):
\[ b = -\frac{5}{9} \]

And substituting \(a\) into (5) to find \(c\):
\[ c = -3 \cdot \frac{5}{9} = -\frac{15}{9} = -\frac{5}{3} \]

### Final Values

Thus, the values of \(a\), \(b\), and \(c\) are:
\[ a = \frac{5}{9}, \quad b = -\frac{5}{9}, \quad c = -\frac{5}{3} \]