In order to find the partial fractions decomposition of the following rational expression, we must first look at the denominator. Now, here we could see that X squared is common between X of the port minus nine exports so we could factor in ah X squared out of extra fourth minus nine experts. So we taken X squared out. We're left with X squared minus not now X squared minus nine Can still be factored X where my science X minus three times x plus ter so we could write Rewrite This is two X plus three over X squared times X minus three X plus three. Now we're ready to start writing our partial fraction decomposition. So if you notice X squared is repeating. So we're gonna write some number A over X plus some number B over X where plus now we ever to non repeating functions, plus some number C over X minus three plus some number D over X plus. OK, now we're ready to try to solve for ABC. Indeed, let's multiply both sides of the equal signed by the common denominator X squared times X squared minus nine and the result will be two X plus three is equal to a X, so we'll multiply. X squared minus nine is X minus three times X plus three plus B times X minus three X plus three. Who are C X squared, X plus three plus de X squared X minus three. Now let's expand. Expand everything on the right. Multiply everything out and then group are like terms, and when you do that, you're left with the identity. Two X plus three is equal to a plus. C plus D. Ex queued waas plus three C was, uh minus was three C minus three D X squared, plus negative nine a axe plus negative nine. Be okay. Now that we approve dollar terms, let's equate the coefficients on both sides of the equal sides. Well, there's no execute on the laugh so we can write a plus B. A plus c plus de is equal is your There's no X Square to Grade B plus three CVE minus three D is equal to zero. There is another two exes on the left, so we could say we were right. That negative nine a will equal to. Likewise, there's a three here. Negative nine b that's what will correspond to that. So we could say that negative nine b is equal to three. OK, now we can saw Let's look at the, uh and be first. Well, if Negative nine A equals two, we divide both sides by negative nine. We're gonna be left with a equals negative, too nice. And for beef, we divide both sides by negative nine. We're gonna get negative three over none, which equals negative 1/3. Now that we have A and B, let's offer, See indeed. Well, let's look at this equation. So if a equals negative 1/3 C plus D will equal 1/3 we add a, which is Nate. We add us not negative. What their A will be negative two nights. So if a is negative two nights C plus D will equal to nice. What about this equation? If B is equal to negative 1/3 and three C minus three D is equal to 1/3. Now let's try solving these two equations on Teasley. Let's divide this equation by three were divided out by three. We're left with C minus. D is equal to one night. And now let's just rewrite this year because we're about to ab use to go c plus d is equal to night. Now we're gonna add we're gonna get rid of our deeds cell for C So this would mean that C plus C that's two C is equal to three night divide both sides by two and you're gonna get C is equal to 3/18 which is ableto warn six. Okay, now that we have a, B and C, let's solve for D Well, if C plus D equals two nights, then they're C plus D is equal to two nights. So that means +16 plus D is equal to two nights. So therefore, D is equal toe d will be equal to one E teeth 29 to minus 16 is 1 18 Now that we have a, B, C and D, we could finish writing are partial fraction decomposition of the rational expression. So a rational expression is two x plus three over X to the fourth minus nine X squared, so that will equal a is negative two nights. So that's gonna equal negative to ninth in your negative to ninth over. X plus B is negative 1/3. So plus negative 1/3 X square plus C is +16 of +16 x minus three plus de is 1 18 Suppose 1/18 X plus three and that is your parcel fraction decomposition of the rational expression in this problem.