Question

What is the simplified form of x minus 5 over x squared minus 3x minus 10 ⋅ x plus 4 over x squared plus x minus 12? (6 points)

Answer 1

Step 1 is to factor everything! 

After factoring you will get:

x-5 over (x+2)(x+5) times x+4 over (x-3)(x+4)

Step 2 is to remove the duplicate factors!
Remove x+4 and x+5

x-5 over (x+2)(x-5) times 1 over x-3 = 1 over x+2 times 1 over x-3 

Step 3 is to multiply the numerators and the denominators.

End result is 1 over (x+2)(x-3)

Answer 2

Answer: Our simplified form will be

$\frac{1}{(x+2)(x-3)}$

Step-by-step explanation:

Since we have given that

$\frac{x-5}{x^2-3x-10}\times \frac{x+4}{x^2+x-12}$

So, first we split the middle term of the denominator ,

$x^2-3x-10=0\\\\=x^2-5x+2x-10\\\\=x(x-5)+2(x-5)\\\\=(x-5)(x+2)$

Similarly,

$x^2+x-12=0\\\\x^2+4x-3x-12=0\\\\x(x+4)-3(x+4)=0\\\\(x+4)(x-3)=0$

Now, we put in our expression above:

$\frac{x-5}{(x-5)(x+2)}\times \frac{x+4}{(x-3)(x+4)}\\\\=\frac{1}{x+2}\times \frac{1}{x-3}\\\\=\frac{1}{(x+2)(x-3)}$

Hence, our simplified form will be

$\frac{1}{(x+2)(x-3)}$