Questions: Simplify. #6) (X^2+4X-5)/(X^2+3X-10) * (X+3)/(X^2+2X-3)

Simplify. #6) (X^2+4X-5)/(X^2+3X-10) * (X+3)/(X^2+2X-3)
Transcript text: Simplify. \#6) $\frac{\mathrm{X}^{2}+4 \mathrm{X}-5}{\mathrm{X}^{2}+3 \mathrm{X}-10} \cdot \frac{\mathrm{X}+3}{\mathrm{X}^{2}+2 \mathrm{X}-3}$
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Solution

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Solution Steps

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

Step 1: Factor the Numerators and Denominators

We start with the expression:

\[ \frac{X^{2}+4X-5}{X^{2}+3X-10} \cdot \frac{X+3}{X^{2}+2X-3} \]

Factoring each part, we find:

  • \( X^{2}+4X-5 = (X - 1)(X + 5) \)
  • \( X^{2}+3X-10 = (X - 2)(X + 5) \)
  • \( X + 3 \) remains as is.
  • \( X^{2}+2X-3 = (X - 1)(X + 3) \)
Step 2: Rewrite the Expression

Substituting the factored forms back into the expression gives us:

\[ \frac{(X - 1)(X + 5)}{(X - 2)(X + 5)} \cdot \frac{X + 3}{(X - 1)(X + 3)} \]

Step 3: Cancel Common Factors

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

  • The factor \( (X + 5) \) cancels.
  • The factor \( (X - 1) \) cancels.
  • The factor \( (X + 3) \) cancels.

This simplifies our expression to:

\[ \frac{1}{X - 2} \]

Final Answer

Thus, the simplified expression is:

\[ \boxed{\frac{1}{X - 2}} \]

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