Expand and collect like terms in ((a1(x - y) + b1)(x - y) + (a2 y + b2) y) / x.

Step-by-step solution

1. 1

   Recognise the nested (Horner) shape. Each half has the form $(\text{coefficient}\cdot t+\text{constant})\cdot t$, which is a quadratic written to minimise multiplications: the first with $t=x-y$, the second with $t=y$. Unfolding one layer at a time is safer than expanding everything at once.

2. 2

   Unfold each half into a quadratic in its own variable.

   $\left(a_1(x-y)+b_1\right)(x-y)=a_1(x-y)^2+b_1(x-y)$

   $\left(a_2y+b_2\right)y=a_2y^2+b_2y$

   so the numerator is $a_1(x-y)^2+b_1(x-y)+a_2y^2+b_2y$.

3. 3

   Expand the square and the linear bracket. Using $(x-y)^2=x^2-2xy+y^2$:

   $a_1(x-y)^2=a_1x^2-2a_1xy+a_1y^2,\qquad b_1(x-y)=b_1x-b_1y$

   Every term of the bracket gets the coefficient, including the middle term $-2a_1xy$ that is easiest to lose.

4. 4

   Collect like terms across the two halves. Only the $y^2$ and the $y$ terms can merge:

   $a_1y^2+a_2y^2=(a_1+a_2)y^2,\qquad -b_1y+b_2y=(b_2-b_1)y$

   The numerator becomes

   $a_1x^2-2a_1xy+(a_1+a_2)y^2+b_1x+(b_2-b_1)y$

5. 5

   Divide term by term if a fully split form is wanted. With $x\neq 0$,

   $a_1x-2a_1y+\frac{(a_1+a_2)y^2}{x}+b_1+\frac{(b_2-b_1)y}{x}$

   Only the terms containing $x$ simplify; the $y^2$ and $y$ terms keep an $x$ in the denominator, so this form is not a polynomial.

6. 6

   Check with numbers. Take $a_1=1,b_1=2,a_2=3,b_2=4,x=5,y=1$. Original numerator: $(1\cdot 4+2)\cdot 4+(3+4)\cdot 1=24+7=31$. Collected form: $25-10+4\cdot 1+10+2\cdot 1=31$ ✓. Dividing by $x=5$ gives $6.2$ either way.