(4x^2 + 15x + 9) / (8x^2 + 10x + 3) divide by (x^2 + 4x) / (2x + 1)
let's see.
4x^2 + 15x +9 factors to 4(x + 3/4)(x + 3).
8x^2 + 10x + 3 factors to 8(x + 3/4)(x + 1/2)
x^2 + 4x factors to x(x + 4)
2x+1 factors to 2(x + 1/2)
so we have
4(x + 3/4)(x + 3) / 8(x + 3/4)(x + 1/2) divide by x(x + 4) / 2(x + 1/2)
there's an x + 3/4 on the top and bottom of the first part, so let's cancel those out
4(x + 3) / 8(x + 1/2) divide by x(x + 4) / 2(x + 1/2)
you can change the divide by to a times if you flip the part on the right
4(x + 3) / 8(x + 1/2) times 2(x + 1/2) / x(x + 4)
multiply the tops together and the bottoms together
8(x + 3)(x + 1/2) / 8x(x + 1/2)(x + 4)
there's an x + 1/2 on the top and bottom, so cancel those out
8(x + 3) / 8x(x + 4)
cancel the 8s out
Solution: (x + 3) / x(x + 4)