Questions: Perform the indicated operation for the expression. List ALL restrictions. 4/(x^2+x-6)+3x/(x-2)-2/(x+3)=(3x^3+16x^2+35x-4)/((x-2)(x+3)) Restrictions: x ≠ 2,-3 ∨ 0^∞

Solution Steps

To solve the given problem, we need to perform the indicated operations on the rational expressions and identify any restrictions on the variable \( x \). The restrictions are values of \( x \) that make any denominator zero.

1. Identify the denominators in the given rational expressions.
2. Determine the values of \( x \) that make any of the denominators zero.
3. List these values as restrictions.

The given rational expressions are: \[ \frac{4}{x^{2}+x-6} + \frac{3x}{x-2} - \frac{2}{x+3} \] The denominators are:

1. \( x^{2} + x - 6 \)
2. \( x - 2 \)
3. \( x + 3 \)

To find the restrictions on \( x \), we need to determine the values that make each denominator equal to zero.

1. For \( x^{2} + x - 6 = 0 \): \[ (x - 2)(x + 3) = 0 \implies x = 2 \text{ or } x = -3 \]
2. For \( x - 2 = 0 \): \[ x = 2 \]
3. For \( x + 3 = 0 \): \[ x = -3 \]

The values that make any denominator zero are \( x = 2 \) and \( x = -3 \). Therefore, the restrictions on \( x \) are: \[ x \neq 2, -3 \]

Final Answer