On this problem, the first thing we have to do is factor the denominator, which is the difference of cubes. So we're going to get X -, 4 * X ^2 + 4 X plus 16. Now we're going to rewrite those as our partial fractions. So we're going to have a constant over the linear a / X -, 4 + a linear over that non reducible quadratic. So BX plus C over X ^2 + 4 X plus 16. We're going to multiply everything through by the common denominator. So we're going to get -2 X squared plus eighteen X + 8 equaling a times the quantity X ^2 + 4 X plus 16 plus the quantity BX plus C * X -, 4. Then we're going to foil out and distribute on the right hand side. So we get AX squared plus 4AX plus 16A plus BX squared -4 BX plus CX -4 C. Now we need to group the coefficients of the squared terms, the X ^2 terms on the left with the coefficients of the X ^2 terms on the right. So -2 equal A+B. We need to group the X terms on the left with the X terms on the right. So 18 equal 4A minus four b + C and then we need to group the terms with no X is on the left with the terms with no X is on the right. So 8 equaling sixteen a -, 4 C When I look at this, these new 3 equations, I want to figure out how to get rid of a variable. We want to do either B or C because there's already B or C missing in each in one of the three equations. So if I get rid of the B, I'm going to take that top equation and multiply it all the way through by 4 -8 equal 4A plus 4B. When I add that with the second equation, we get 10 equaling 8A plus C. Now if we look at that third equation, it's all divisible by 4, so we get 2 equaling four a -, C If we add that directly with that 10 equaling 8A plus C, we get 12 equal 12A or a is one. Then I'm going to substitute the A equal 1 into one of the equations. I use the two equal four a -, C so 2 equal 4 * 1 -, C negative 2 equal negative C or C is 2. And once I get my A and my C, I'm going to go back to the original and find an equation with AB in it. I used the top one -2 equal a + b or -2 equal 1 + b, so B is -3. So our final answer is 1 / X -, 4 + -3 X plus 2 / X ^2 + 4 X plus 16.