Express (15x^2+5x+2)/(2-x)(1+2x^2) in partial fractions.

To express the given rational expression in partial fractions, we need to decompose it into simpler fractions.

First, let's factor the denominators:
(2 - x) can be factored into (1 - x) after changing the order.
(1 + 2x^2) is a quadratic expression in standard form.

Now, let's express the given rational expression as the sum of two fractions:
(15x^2 + 5x + 2) / [(2 - x)(1 + 2x^2)] = A / (2 - x) + (Bx + C) / (1 + 2x^2)

Next, let's find the values of A, B, and C.

First, let's eliminate (2 - x) from the first fraction. We can do this by multiplying both sides of the equation by (2 - x):
(15x^2 + 5x + 2) = A + (Bx + C)(2 - x)

Expanding the right side of the equation, we get:
15x^2 + 5x + 2 = A + (2B - Bx + 2C - Cx)

To determine the values of A, B, and C, we can compare the coefficients of like terms on both sides.

For the x^2 term:
0x^2 = -B + 15x^2
=> B = -15

For the x term:
5x = -Cx - Bx
=> 5x = -(C + B)x
Comparing coefficients, we get:
C + B = 0
C - 15 = 0
=> C = 15

For the constant term:
2 = A + 2C
=> A = 2 - 2C
=> A = 2 - 2(15)
=> A = 2 - 30
=> A = -28

Now that we have found the values of A, B, and C, we can rewrite the rational expression using the partial fraction decomposition:

(15x^2 + 5x + 2) / [(2 - x)(1 + 2x^2)] = -28 / (2 - x) + (-15x + 15) / (1 + 2x^2)

So, the given rational expression expressed in partial fractions is:
(15x^2 + 5x + 2) / [(2 - x)(1 + 2x^2)] = -28 / (2 - x) + (-15x + 15) / (1 + 2x^2)