If frac{2 x^2+5 x+6}{(x+2)^3}=frac{a}{x+2}+fr | vedclass

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(C) Given the partial fraction decomposition: $\frac{2 x^2+5 x+6}{(x+2)^3}=\frac{a}{x+2}+\frac{b}{(x+2)^2}+\frac{c}{(x+2)^3}$ Multiply both sides by $

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-$10$

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(C) Given the partial fraction decomposition: $\frac{2 x^2+5 x+6}{(x+2)^3}=\frac{a}{x+2}+\frac{b}{(x+2)^2}+\frac{c}{(x+2)^3}$ Multiply both sides by $(x+2)^3$: $2 x^2+5 x+6 = a(x+2)^2 + b(x+2) + c$ Let $x+2 = t$,then $x = t-2$. Substituting this: $2(t-2)^2 + 5(t-2) + 6 = a t^2 + b t + c$ $2(t^2 - 4t + 4) + 5t - 10 + 6 = a t^2 + b t + c$ $2t^2 - 8t + 8 + 5t - 4 = a t^2 + b t + c$ $2t^2 - 3t + 4 = a t^2 + b t + c$ Comparing coefficients: $a = 2, b = -3, c = 4$ Now,calculate $a \cdot b + b \cdot c + c \cdot a$: $a \cdot b = 2 \cdot (-3) = -6$ $b \cdot c = (-3) \cdot 4 = -12$ $c \cdot a = 4 \cdot 2 = 8$ Sum $= -6 - 12 + 8 = -10$

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

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