Need explanation on one part

Question

Use the following expression to answer the questions below:

(1/x+3 + 6/x^2+4x+3)*x+3/x+1

a. Simplify the expression down to one rational expression.

b. Explain the steps you used to do the simplification.

c. Are there any values x cannot be? Explain.

d. Find the values of x that make the expression equal to 0. Explain the steps you used to find these values.

I need someone to explain how to do questions c and d.

Answer 1

Factoring the denominator we see that the denominators all consist of the expressions "x + 3" and "x + 1".

Based on the fact that division by zero is undefined, x ≠ -3 and x ≠ -1

Set the expression equal to zero then multiply it out (distribute the "x + 3" to the two numerators and the "x + 1" to the two denominators to get:

Since the denominator cannot contribute to the expression being equal to zero (it cannot BE zero, and if it is anything else, that will not help the value to be zero), then we can just ignore the denominator. So the expression can equal zero ONLY when the numerator is equal to zero:

x² + 10x + 21 = 0

(x + 7)(x + 3) = 0

x = -7 and x = -3

However, we have already stated that x ≠ -3 therefore, the value of "x" that makes the expression equal to zero is x = -7

Answer 2

[1 / (x+3) + 6 / (x²+4x+3)] [(x+3)/(x+1)]

= [1 / (x+3) + 6 / ((x+3)(x+1))] [ (x+3)/(x+1) ]

= 1 / (x+1) + 6 / (x+1)² = (x+1) / (x+1)² + 6 / (x+1)² = (x + 7) / (x + 1)²

c. Since x+3 and x+1 appear in denominators, x cannot be -3 or -1.

d. Setting the simplified expression equal to 0, we see that x+7 = 0. So x = -7.