QUESTION 15 What value(s) are not allowable replacements for the variable $$ x $$ ? Do not solve the equation $$ \frac{2 x}{x+2}-\frac{3 x}{x+5}=\frac{4}{x^{2}+7 x+10} $$

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To determine the **non-allowable** values for $$ x $$ in the equation:

$$
\frac{2x}{x + 2} - \frac{3x}{x + 5} = \frac{4}{x^2 + 7x + 10}
$$

we need to identify the values of $$ x $$ that would make any denominator equal to zero, as division by zero is undefined.

1. **First Denominator:** $$ x + 2 $$
   $$
   x + 2 \neq 0 \implies x \neq -2
   $$

2. **Second Denominator:** $$ x + 5 $$
   $$
   x + 5 \neq 0 \implies x \neq -5
   $$

3. **Third Denominator:** $$ x^2 + 7x + 10 $$
   Factor the quadratic:
   $$
   x^2 + 7x + 10 = (x + 2)(x + 5)
   $$
   $$
   (x + 2)(x + 5) \neq 0 \implies x \neq -2 \text{ and } x \neq -5
   $$

Combining these results, the **non-allowable** values for $$ x $$ are:

$$
\boxed{\left\{\, -5,\; -2\,\right\}}
$$
The values that are not allowable for $$ x $$ are $$ -5 $$ and $$ -2 $$.

QUESTION 15 What value(s) are not allowable replacements for the variable \( x \) ? Do not solve the equation \( \frac{2 x}{x+2}-\frac{3 x}{x+5}=\frac{4}{x^{2}+7 x+10} \)

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