$$ \frac { b } { x - a } = \frac { x } { x ^ { 2 } + a x + a } $$

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It seems that the calculations are too complex for direct evaluation. Let's break down the steps manually instead of relying on automated calculations.

### Step 1: Identify coefficients

From the quadratic equation we derived:

$$
(b - 1)x^2 + (ab + a)x + ba = 0
$$

We have:
- $$A = b - 1$$
- $$B = ab + a$$
- $$C = ba$$

### Step 2: Calculate the discriminant

The discriminant $$D$$ is given by:

$$
D = B^2 - 4AC
$$

Substituting the values of $$A$$, $$B$$, and $$C$$:

$$
D = (ab + a)^2 - 4(b - 1)(ba)
$$

### Step 3: Expand the discriminant

Let's expand $$D$$:

1. Calculate $$B^2$$:
   $$
   (ab + a)^2 = a^2b^2 + 2a^2b + a^2
   $$

2. Calculate $$4AC$$:
   $$
   4(b - 1)(ba) = 4b^2a - 4ba
   $$

Now, substituting these into the discriminant:

$$
D = a^2b^2 + 2a^2b + a^2 - (4b^2a - 4ba)
$$

This simplifies to:

$$
D = a^2b^2 + 2a^2b + a^2 - 4b^2a + 4ba
$$

### Step 4: Solve for $$x$$

Now, we can use the quadratic formula:

$$
x = \frac{-B \pm \sqrt{D}}{2A}
$$

Substituting $$A$$, $$B$$, and $$D$$ into the formula will give us the solutions for $$x$$.

### Final Expression for $$x$$

The final expression for $$x$$ will be:

$$
x = \frac{-(ab + a) \pm \sqrt{D}}{2(b - 1)}
$$

This gives us the solutions for $$x$$ in terms of $$a$$ and $$b$$. If you have specific values for $$a$$ and $$b$$, we can substitute them in to find numerical solutions. Would you like to proceed with specific values?
To solve the equation $$ \frac{b}{x - a} = \frac{x}{x^2 + ax + a} $$, follow these steps: 1. **Cross-multiply** to eliminate the fractions: $$ b(x^2 + ax + a) = x(x - a) $$ 2. **Expand both sides**: $$ bx^2 + abx + ba = x^2 - ax $$ 3. **Set the equation to zero**: $$ bx^2 + abx + ba - x^2 + ax = 0 $$ Simplifies to: $$ (b - 1)x^2 + (ab + a)x + ba = 0 $$ 4. **Use the quadratic formula** to find $$x$$: $$ x = \frac{-(ab + a) \pm \sqrt{(ab + a)^2 - 4(b - 1)(ba)}}{2(b - 1)} $$ This gives the solutions for $$x$$ in terms of $$a$$ and $$b$$.

\( \frac { b } { x - a } = \frac { x } { x ^ { 2 } + a x + a } \)

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