With



the singular ODE can be written as
![\displaystyle\sum_(n\ge0)\bigg[2(n+r)(n+r-1)+3(n+r)-1\bigg]a_nx^(n+r)+\sum_(n\ge0)\bigg[-2(n+r)(n+r-1)-3(n+r)\bigg]a_nx^(n+r-1)=0](https://img.qammunity.org/2018/formulas/mathematics/college/t4ijrrsbxhibcefz8y927r4b8ojhd5ad1w.png)
The first term of the second series admits the indicial equation. When

, we have

Factoring reveals two distinct roots at

(in your case, swap

and

before submitting).
Next, shift the index of the first sum so that it starts at

by replacing

, then consolidate the sums to get
![\displaystyle\sum_(n\ge1)\bigg[2(n+r-1)(n+r-2)+3(n+r-1)-1-2(n+r)(n+r-1)-3(n+r)\bigg]a_nx^(n+r-1)](https://img.qammunity.org/2018/formulas/mathematics/college/spwdzz5dvzhsd7bh1q37u7w0k46h3daara.png)
![=\displaystyle x^r\sum_(n\ge1)\bigg[-4n-4r\bigg]a_nx^(n-1)](https://img.qammunity.org/2018/formulas/mathematics/college/p5j4ikivxc4f0czlqpmuuya5u4ql999obu.png)
Setting

, we then have this as


However I don't see the connection to the given answer... It seems some information is missing, specifically about how the coefficients

are related.