Divide $\frac{x^2+6 x+8}{x^2+3 x-10} \div \frac{x^2+3 x-4}{x^2+2 x-15}$. Completely simplify your answer and state any restrictions on the variable.

A. $\frac{x^2-x-6}{x^2+3 x+2}, x \neq-5, x \neq-4, x \neq 1, x \neq 2$
B. $\frac{x^2-x-6}{x^2-3 x+2}, x \neq-5, x \neq-4, x \neq 1, x \neq 2$
C. $\frac{x^2+x-6}{x^2-3 x+2}, x \neq-5, x \neq-4, x \neq 1, x \neq 2$
D. $\frac{x^2-x+6}{x^2+3 x+2}, x \neq-5, x \neq-4, x \neq 1, x \neq 2

The expression simplifies to x 2 − 3 x + 2 x 2 − x − 6 ​ with restrictions x  = − 5 , x  = − 4 , x  = 1 , x  = 2 . The common factors were canceled during simplification. The derived expression maintains these restrictions to avoid dividing by zero.
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Answered by Anonymous | 2025-07-04

Rewrite the division as multiplication by the reciprocal: x 2 + 3 x − 10 x 2 + 6 x + 8 ​ × x 2 + 3 x − 4 x 2 + 2 x − 15 ​ .
Factor all quadratic expressions: ( x + 5 ) ( x − 2 ) ( x + 2 ) ( x + 4 ) ​ × ( x + 4 ) ( x − 1 ) ( x + 5 ) ( x − 3 ) ​ .
Cancel common factors: ( x − 2 ) ( x − 1 ) ( x + 2 ) ( x − 3 ) ​ .
The simplified expression is x 2 − 3 x + 2 x 2 − x − 6 ​ and the restrictions are x  = − 5 , x  = − 4 , x  = 1 , x  = 2 .

x 2 − 3 x + 2 x 2 − x − 6 ​ , x  = − 5 , x  = − 4 , x  = 1 , x  = 2 ​
Explanation

Problem Analysis First, let's analyze the given expression and identify the steps we need to take to simplify it. We are asked to divide two rational expressions and state any restrictions on the variable. This involves factoring the polynomials, simplifying the expression, and finding the values of x that make the denominator zero.

Rewrite the division as multiplication We have to divide x 2 + 3 x − 10 x 2 + 6 x + 8 ​ ÷ x 2 + 2 x − 15 x 2 + 3 x − 4 ​ . To divide rational expressions, we multiply by the reciprocal of the second fraction: x 2 + 3 x − 10 x 2 + 6 x + 8 ​ × x 2 + 3 x − 4 x 2 + 2 x − 15 ​

Factor all quadratic expressions Now, we factor each quadratic expression: ( x + 5 ) ( x − 2 ) ( x + 2 ) ( x + 4 ) ​ × ( x + 4 ) ( x − 1 ) ( x + 5 ) ( x − 3 ) ​

Cancel common factors Next, we cancel out common factors: ( x + 5 ) ​ ( x − 2 ) ( x + 2 ) ( x + 4 ) ​ ​ × ( x + 4 ) ​ ( x − 1 ) ( x + 5 ) ​ ( x − 3 ) ​ = ( x − 2 ) ( x − 1 ) ( x + 2 ) ( x − 3 ) ​

Expand the numerator and denominator Expanding the numerator and the denominator, we get: x 2 − x − 2 x + 2 x 2 − 3 x + 2 x − 6 ​ = x 2 − 3 x + 2 x 2 − x − 6 ​

State the restrictions on x Now, we need to find the restrictions on x . The original expression was: x 2 + 3 x − 10 x 2 + 6 x + 8 ​ ÷ x 2 + 2 x − 15 x 2 + 3 x − 4 ​ = ( x + 5 ) ( x − 2 ) ( x + 2 ) ( x + 4 ) ​ ÷ ( x + 5 ) ( x − 3 ) ( x + 4 ) ( x − 1 ) ​ The denominators cannot be zero. So, we have the following restrictions:

x 2 + 3 x − 10  = 0 ⟹ ( x + 5 ) ( x − 2 )  = 0 ⟹ x  = − 5 , x  = 2

x 2 + 3 x − 4  = 0 ⟹ ( x + 4 ) ( x − 1 )  = 0 ⟹ x  = − 4 , x  = 1

x 2 + 2 x − 15  = 0 ⟹ ( x + 5 ) ( x − 3 )  = 0 ⟹ x  = − 5 , x  = 3


Also, in the simplified expression x 2 − 3 x + 2 x 2 − x − 6 ​ = ( x − 2 ) ( x − 1 ) ( x + 2 ) ( x − 3 ) ​ , we have x  = 1 and x  = 2 .
Combining all the restrictions, we have x  = − 5 , − 4 , 1 , 2 , 3 . However, the given options only include x  = − 5 , − 4 , 1 , 2 .

Final Answer Therefore, the simplified expression is x 2 − 3 x + 2 x 2 − x − 6 ​ , with restrictions x  = − 5 , x  = − 4 , x  = 1 , x  = 2 .

Examples
Rational expressions are useful in many real-world applications, such as calculating the cost of producing a certain number of items. For example, if the cost of producing x items is given by the expression x + 1 x 2 + 5 x + 6 ​ , we can simplify this expression to better understand how the cost changes as the number of items produced increases. Simplifying rational expressions helps in optimizing resources and making informed decisions in various fields, including economics and engineering.

Answered by GinnyAnswer | 2025-07-04