To solve the problem shown in the image, we need to subtract two rational expressions. Here’s the step-by-step approach:

Step 1: Factor the Denominators

The denominators need to be factored to simplify the subtraction.

• First denominator: $x^2 - 4$ $x^2 - 4 = (x - 2)(x + 2)$

• Second denominator: $x^2 - 10x + 16$ $x^2 - 10x + 16 = (x - 2)(x - 8)$

Step 2: Combine the Rational Expressions

Now that both denominators are factored, we want to subtract: $\frac{2x + 4}{(x - 2)(x + 2)} - \frac{2x - 9}{(x - 2)(x - 8)}$

To subtract these, we need a common denominator, which is the product of all unique factors from both denominators: $(x - 2)(x + 2)(x - 8)$

Thus, the new expressions are:

• First term: $\frac{2x + 4}{(x - 2)(x + 2)} \times \frac{x - 8}{x - 8} = \frac{(2x + 4)(x - 8)}{(x - 2)(x + 2)(x - 8)}$

• Second term: $\frac{2x - 9}{(x - 2)(x - 8)} \times \frac{x + 2}{x + 2} = \frac{(2x - 9)(x + 2)}{(x - 2)(x + 2)(x - 8)}$

Step 3: Simplify the Numerators

Now, expand both numerators:

• First numerator: $(2x + 4)(x - 8) = 2x(x - 8) + 4(x - 8) = 2x^2 - 16x + 4x - 32 = 2x^2 - 12x - 32$

• Second numerator: $(2x - 9)(x + 2) = 2x(x + 2) - 9(x + 2) = 2x^2 + 4x - 9x - 18 = 2x^2 - 5x - 18$

Step 4: Subtract the Numerators

Now, subtract the numerators: $(2x^2 - 12x - 32) - (2x^2 - 5x - 18) = 2x^2 - 12x - 32 - 2x^2 + 5x + 18$ This simplifies to: $-12x + 5x - 32 + 18 = -7x - 14$

Step 5: Final Expression

Thus, the final expression is: $\frac{-7x - 14}{(x - 2)(x + 2)(x - 8)}$

Step 6: Simplify the Numerator (if possible)

We can factor the numerator: $-7x - 14 = -7(x + 2)$

Thus, the simplified result is: $\frac{-7(x + 2)}{(x - 2)(x + 2)(x - 8)}$

We can cancel $(x + 2)$ from both the numerator and denominator: $\frac{-7}{(x - 2)(x - 8)}$

Final Answer:

The simplified result of the given operation is: $\frac{-7}{(x - 2)(x - 8)}$

Would you like more details on any of the steps, or do you have any questions?

Here are five related questions:

1. How do you find a common denominator for rational expressions?
2. What is the method to factor quadratic expressions?
3. How do you subtract polynomials step by step?
4. What is the importance of canceling common factors in rational expressions?
5. How do you expand and simplify binomial products?

Tip: Always remember to look for common factors in the numerator and denominator to simplify rational expressions efficiently.