Putting the chapter together
By now you've met every move in this function separately. It loads an element at a fixed index, loads another at a variable index with lwzx, grabs a neighbor off the computed base, multiplies two loaded values together register-to-register, and subtracts at the end. So nothing here is unfamiliar. The earlier lessons have just been bolted together into one routine.
The efficiency worth catching is that the variable index only gets scaled once, by slwi. After that, the same offset does two jobs. It drives the indexed lwzx, and it also locates the neighbor, whose base falls out of an add before a displacement load finishes the read. The element at index 0 never goes near that path, since a plain lwz 0(r3) reaches it directly.
mix(q, j) blends the first element with the product of two neighbors:
slwi r0, r4, 2 # j * 4
lwz r5, 0(r3) # q[0]
add r4, r3, r0 # r4 = &q[j]
lwzx r3, r3, r0 # q[j]
lwz r0, 4(r4) # q[j+1] (4 / 4 = 1 element past q[j])
mullw r0, r3, r0 # q[j] * q[j+1]
add r3, r5, r0 # q[0] + q[j]*q[j+1]
blr
Here the single slwi is what powers both the lwzx and the add-built base, and because the multiply takes two loaded values it comes out as mullw rather than mulli, with the fixed element joining only at the close. Your target is built from the same parts, but it puts the multiply and the subtract in different places. So work each loaded register forward from its load to whatever consumes it, recover the indices from the displacements, and the expression reassembles itself.
Your task
Write capstone, taking an int* and an int, to reproduce the assembly above.