3 Express the following proper fractions as partial fractions. a \( \frac{2 x^{2}-3 x+2}{x\left(x^{2}+1\right)} \) b \( \frac{3 x^{2}+4 x+17}{(2 x+1)\left(x^{2}+5\right)} \) c \( \frac{2 x^{2}-6 x-9}{(3 x+5)\left(2 x^{2}+1\right)} \) d \( -\frac{6 x^{2}-21 x+50}{(3 x-5)\left(2 x^{2}+5\right)} \)

To express the given proper fractions as partial fractions, we break them down in a way that simplifies each fraction into a sum of simpler fractions. Here’s how we tackle each part: a) For \( \frac{2 x^{2}-3 x+2}{x\left(x^{2}+1\right)} \): We express this as \( \frac{A}{x} + \frac{Bx+C}{x^{2}+1} \). Here, \( A, B, C \) are constants we need to determine. Multiplying through by \( x(x^{2}+1) \) and equating coefficients gives us the desired values for \( A, B, \) and \( C \). b) For \( \frac{3 x^{2}+4 x+17}{(2 x+1)\left(x^{2}+5\right)} \): We write it as \( \frac{A}{2x+1} + \frac{Bx+C}{x^{2}+5} \). Similar to part a, we’ll multiply through by the denominator, group like terms, and solve for \( A, B, \) and \( C \). c) For \( \frac{2 x^{2}-6 x-9}{(3 x+5)\left(2 x^{2}+1\right)} \): Here, we have \( \frac{A}{3x+5} + \frac{Bx+C}{2x^{2}+1} \). Again, multiply through, set up the equations for coefficients, and find \( A, B, \) and \( C \). d) For \( -\frac{6 x^{2}-21 x+50}{(3 x-5)\left(2 x^{2}+5\right)} \): We can represent this as \( \frac{A}{3x-5} + \frac{Bx+C}{2x^{2}+5} \). After multiplying through, we’ll isolate coefficients leading to values for \( A, B, \) and \( C \). In each case, once you find \( A, B, \) and \( C \), substitute them back into the partial fraction components to complete the expression.