Express 2x^3-9/x^3+6x^2+8x as partial fractions

(2x^3-9)/(x^3+6x^2+8x)

= (2x^3-9) / (x)(x+2)(x+4)
= 2 - (9/8)/x + (25/4)/(x+2) - (137/8)/(x+4)

To express the given rational function, 2x^3-9/x^3+6x^2+8x, as partial fractions, follow these steps:

Step 1: Factor the denominator.
The denominator, x^3 + 6x^2 + 8x, can be factored as follows:
x(x^2 + 6x + 8) = x(x + 2)(x + 4).

Step 2: Write the partial fractions.
Since the degree of the numerator (3) is greater than the degree of the denominator (1), we need to use proper fractions. We write the partial fractions as:
2x^3 - 9/x(x + 2)(x + 4) = A/x + B/(x + 2) + C/(x + 4).

Step 3: Find the values of A, B, and C.
To find the values of A, B, and C, we need to find the common denominator. This is achieved by multiplying each fraction by the respective denominator of the other fractions:
2x^3 - 9 = A(x + 2)(x + 4) + Bx(x + 4) + Cx(x + 2).

Step 4: Simplify and solve for A, B, and C.
Expand the equation and collect like terms:
2x^3 - 9 = (A + B + C)x^2 + (6A + 3B + 2C)x + (8A + 4B).

By equating the coefficients on both sides of the equation, we can form a system of equations:
Coefficients of x^2: A + B + C = 2
Coefficients of x: 6A + 3B + 2C = 0
Coefficients of the constant term: 8A + 4B = -9

Solve this system of equations to find the values of A, B, and C.

Step 5: Add the partial fractions.
Once you have determined the values of A, B, and C, rewrite the rational function by plugging in these values into the partial fractions expression:
2x^3 - 9/x^3 + 6x^2 + 8x = A/x + B/(x + 2) + C/(x + 4).

Now you have expressed the given rational function as partial fractions.