Let's simplify the given expression step-by-step.
The given expression is:
[tex]\[
\frac{x^2 + 6x - 7}{x^2 - 36} \cdot \frac{x^2 - 2x - 24}{2x^2 + 8x - 42}
\][/tex]
1. Factorize each expression in the numerator and the denominator:
- Factorizing [tex]\( x^2 + 6x - 7 \)[/tex]:
[tex]\[
x^2 + 6x - 7 = (x + 7)(x - 1)
\][/tex]
- Factorizing [tex]\( x^2 - 36 \)[/tex] (Difference of squares):
[tex]\[
x^2 - 36 = (x + 6)(x - 6)
\][/tex]
- Factorizing [tex]\( x^2 - 2x - 24 \)[/tex]:
[tex]\[
x^2 - 2x - 24 = (x - 6)(x + 4)
\][/tex]
- Factorizing [tex]\( 2x^2 + 8x - 42 \)[/tex]:
First, factor out the common factor of 2:
[tex]\[
2x^2 + 8x - 42 = 2(x^2 + 4x - 21)
\][/tex]
Now factorize [tex]\( x^2 + 4x - 21 \)[/tex]:
[tex]\[
x^2 + 4x - 21 = (x + 7)(x - 3)
\][/tex]
Therefore,
[tex]\[
2x^2 + 8x - 42 = 2(x + 7)(x - 3)
\][/tex]
2. Rewrite the original expression with the factors:
[tex]\[
\frac{(x + 7)(x - 1)}{(x + 6)(x - 6)} \cdot \frac{(x - 6)(x + 4)}{2(x + 7)(x - 3)}
\][/tex]
3. Simplify by canceling common factors:
- [tex]\( (x + 7) \)[/tex] in the numerator and denominator
- [tex]\( (x - 6) \)[/tex] in the numerator and denominator
The expression simplifies to:
[tex]\[
\frac{(x - 1)(x + 4)}{(x + 6)} \cdot \frac{1}{2(x - 3)}
\][/tex]
This can be simplified further by multiplying:
[tex]\[
\frac{(x - 1)(x + 4)}{2(x + 6)(x - 3)}
\][/tex]
So, the simplified expression is:
[tex]\[
\frac{(x - 1)(x + 4)}{2(x + 6)(x - 3)}
\][/tex]
