Questions: Simplify: (x^2-3x-4)/(x^2-6x+8) * (x^2-7x+10)/(x^2+5x+4)

Transcript text: Simplify: $\frac{x^{2}-3 x-4}{x^{2}-6 x+8} \cdot \frac{x^{2}-7 x+10}{x^{2}+5 x+4}$

Solution

Solution Steps

To simplify the given expression, we need to factor each polynomial in the numerators and denominators. Once factored, we can cancel out any common factors between the numerators and denominators.

Step 1: Factor the Expressions

We start by factoring each polynomial in the expression:

$ x^2 - 3x - 4 = (x - 4)(x + 1) $
$ x^2 - 6x + 8 = (x - 4)(x - 2) $
$ x^2 - 7x + 10 = (x - 5)(x - 2) $
$ x^2 + 5x + 4 = (x + 1)(x + 4) $

Step 2: Rewrite the Expression

Substituting the factored forms into the original expression, we have:

\[ \frac{(x - 4)(x + 1)}{(x - 4)(x - 2)} \cdot \frac{(x - 5)(x - 2)}{(x + 1)(x + 4)} \]

Step 3: Cancel Common Factors

Next, we can cancel the common factors in the numerator and denominator:

The factor $ (x - 4) $ cancels out.
The factor $ (x + 1) $ cancels out.

This simplifies our expression to:

\[ \frac{(x - 5)}{(x - 2)(x + 4)} \]

Final Answer

The simplified expression is:

\[ \boxed{\frac{x - 5}{(x - 2)(x + 4)}} \]

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