Here are my answers. Can you check if I got the right answers? Thank you!

(x-4)/(x^2-3x-4) = x+1
(x^3-8)/(x-2)=x^2+2x+4
(5-x)/(x^2-25)=-x-5
(x^2-4x-32)/(x^2-16)=(x-8)/(x-4)

first one should be 1/(x+1)

second is good
third: -1/(x+5)
4th is ok

Have you not been taught to place restrictions after your answers for variable denominators?

I can help you check your answers for the given equations. Let's go through each equation one by one:

1. (x-4)/(x^2-3x-4) = x+1
To solve this equation, we need to simplify the left side of the equation first by factoring the denominator:
x^2 - 3x - 4 = (x-4)(x+1)
So the equation becomes: (x-4)/((x-4)(x+1)) = x+1

Now, we can cancel out (x-4) on both sides of the equation:
1/(x+1) = x+1
To isolate the variable, we can multiply both sides by (x+1):
1 = (x+1)^2
Expanding the right side gives us:
1 = x^2 + 2x + 1

So the correct equation after simplification is: x^2 + 2x = 0
The solution to this equation is x = 0 and x = -2.

Therefore, the correct answer for the first equation is not 1/(x+1). The correct answer is x = 0 and x = -2.

2. (x^3-8)/(x-2) = x^2 + 2x + 4
To solve this equation, we can factor the numerator using the difference of cubes formula:
x^3 - 8 = (x-2)(x^2 + 2x + 4)
So the equation becomes: (x-2)(x^2 + 2x + 4)/(x-2) = x^2 + 2x + 4

We can cancel out (x-2) on both sides of the equation:
x^2 + 2x + 4 = x^2 + 2x + 4
The equation is already simplified.

Therefore, the correct answer for the second equation is x^2 + 2x + 4.

3. (5-x)/(x^2-25) = -x-5
To solve this equation, we can factor the denominator:
x^2 - 25 = (x-5)(x+5)
So the equation becomes: (5-x)/((x-5)(x+5)) = -x-5

Now we can cancel out (5-x) on both sides of the equation:
1/((x-5)(x+5)) = -1
To get rid of the reciprocal on the left side, we can multiply both sides by (x-5)(x+5):
(x-5)(x+5) = -1

Expanding the left side of the equation gives us:
x^2 - 25 = -1
Adding 1 to both sides:
x^2 = 24

Taking the square root of both sides gives us:
x = ±√24
Simplifying further:
x = ±√(2*2*2*3)
x = ±2√6

Therefore, the correct answer for the third equation is x = ±2√6.

4. (x^2-4x-32)/(x^2-16) = (x-8)/(x-4)
To solve this equation, let's first factor both the numerator and denominator:
x^2 - 4x - 32 = (x-8)(x+4)
x^2 - 16 = (x-4)(x+4)

The equation becomes: (x-8)(x+4)/(x-4)(x+4) = (x-8)/(x-4)

Now, we can cancel out (x-8) on both sides of the equation:
1/(x+4) = 1
To isolate the variable, we can multiply both sides by (x+4):
1 = x+4
Subtracting 4 from both sides:
-3 = x

Therefore, the correct answer for the fourth equation is x = -3.

It is important to note that some of the equations have restrictions due to the denominators. For example, in the third equation (5-x)/(x^2-25) = -x-5, the denominator (x^2-25) cannot be equal to zero. So we need to exclude the values of x that make the denominator zero, which are x = -5 and x = 5. When stating the final answer, we should mention that the solution is x = ±2√6, excluding x = -5 and x = 5.

I hope this explanation helps! Let me know if you have any further questions.