Final answer:
To simplify the given expression, we need to combine the two fractions into a single fraction by finding a common denominator. The simplified expression is (A+Ba)x² + (Bb+Cx) + Ac² + Cb / (ax+b)(x²+c²).
Step-by-step explanation:
To simplify the given expression, we need to combine the two fractions into a single fraction. To do this, we need to find a common denominator. In this case, the common denominator is (ax+b)(x²+c²). We then multiply the first fraction by (x²+c²)/(x²+c²) and the second fraction by (ax+b)/(ax+b). This gives us (A(x²+c²) + (Bx+C)(ax+b))/(ax+b)(x²+c²).
We can further simplify the numerator by expanding the terms and simplifying:
Ax² + Ac² + Bax² + Bbx + Cax + Cb = (A+Ba)x² + (Bb+Cx) + Ac² + Cb.
Therefore, the simplified expression is (A+Ba)x² + (Bb+Cx) + Ac² + Cb / (ax+b)(x²+c²).