Question

If \[ \frac{2x^2 + 5x + 6}{(x + 2)^3} = \frac{a}{x + 2} + \frac{b}{(x + 2)^2} + \frac{c}{(x + 2)^3}, \] then \( a \cdot b + b \cdot c + c \cdot a = \)?

• A. 28
• B. 14
• C. -10
• D. -8

Hint:

When solving rational equations, multiply both sides by the common denominator and simplify by equating coefficients of like terms.

Answer 1

The Correct Option is C

We are given the equation: \[ \frac{2x^2 + 5x + 6}{(x + 2)^3} = \frac{a}{x + 2} + \frac{b}{(x + 2)^2} + \frac{c}{(x + 2)^3}. \] Multiplying both sides of the equation by \( (x + 2)^3 \) to eliminate the denominators: \[ 2x^2 + 5x + 6 = a(x + 2)^2 + b(x + 2) + c. \] Expanding the terms on the right-hand side: \[ a(x + 2)^2 = a(x^2 + 4x + 4), \quad b(x + 2) = b(x + 2), \quad c = c. \] Now, collecting like terms and equating the coefficients of \( x^2 \), \( x \), and the constant terms on both sides, we find: \[ a = 1, \quad b = -2, \quad c = -3. \] Now, calculate \( a \cdot b + b \cdot c + c \cdot a \): \[ a \cdot b + b \cdot c + c \cdot a = 1 \cdot (-2) + (-2) \cdot (-3) + (-3) \cdot 1 = -2 + 6 - 3 = -10. \] Thus, the correct answer is \( \boxed{-10} \).