For this problem we need to have a / X + b / X ^2. Because X ^2 is a linear term, we need every power up until whatever the power that the linear term is 2. But then we also are going to have acx plus d / X ^2 + 4. The reason for this is the X ^2 + 4 is factored as much as it can be and it's quadratic. So above the quadratic has got to be a linear. So now when we multiply through by the common denominator, we're going to get X + 7 equaling axx squared plus 4 + b * X ^2 + 4 plus X ^2 times the CX plus D So if I have X + 7, that's going to equal AX cubed plus 4AX plus BX squared plus 4B plus CX cubed plus DX squared. So the cube terms AX cubed and CX cubed have got to equal the cubed terms on the other side. So we know that 0 = a + C The X squared terms on the one side have to equal the X ^2 terms on the other. So zero is going to equal b + d The X term on one side has got to equal the X term on the other. So the coefficient of one is going to equal 4A and then the constant has to equal the constant. Now a way to check it is to count how many terms you have 123456 and just make sure we had six here so that we didn't lose any. Well, we can see that if one equal 4A, a is 1/4 and if A is 1/4 and we know that A + C = 0, then C has got to be the opposite of A or negative 1/4. We can see B is going to be 7 fourths and if B + D has to equal 0 then D's got to be the opposite or -7 fourths. So we're going to have one over 4X plus seven over four, X ^2 + -1 fourth, X -7 fourths all over X ^2 + 4. Now we can make that into a proper fraction by multiplying the top and the bottom each by 4. So we'd get one over four X + 7 / 4 X squared plus negative X -, 7 / 4 times the quantity X ^2 + 4. So that looks like that's going to be answer D1 over four X + 7 / 4 X squared plus negative X - 7 / 4 * X ^2 + 4.