The answer is the option d, which is: d)
The explanation for this problem is shown below:
1. Smplify the denominator and rewrite the numerator in this form:
2. Multiply the denominator and the numerator by the conjugated and simplify the expression, as following:
3. As you can see, you obtain the expression shown in the option mentioned above.
Answer: Any real number x as long as and
In other words, anything but 0 or -2/3 is valid.
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Explanation:
Set the denominator equal to zero and solve for x
2(3x^2 + 2x) = 0
3x^2 + 2x = 0
x(3x + 2) = 0
x = 0 or 3x+2 = 0 .... zero product property
x = 0 or 3x = -2
x = 0 or x = -2/3
If either x = 0 or x = -2/3, then the denominator 2(3x^2 + 2x) will be zero. But recall that we cannot have zero in the denominator. Dividing by zero is not allowed. The expression is undefined when we divide by zero.
Therefore, we must exclude x = 0 and x = -2/3 from the domain. Any other real number is valid as an x input.