Question

simplify the following rational expressions

5x x blank
x2^+5x+6 + x+3 = x2^+5x+6

answer the blank

Answer 1

To simplify the rational expression

\[
\frac{5x}{x^2 + 5x + 6} + \frac{x}{x + 3} = \frac{x^2 + 5x + 6}{\text{blank}},
\]

we first factor the expression in the denominator \(x^2 + 5x + 6\).

The expression \(x^2 + 5x + 6\) can be factored as:

\[
x^2 + 5x + 6 = (x + 2)(x + 3).
\]

Thus, we rewrite the equation as:

\[
\frac{5x}{(x + 2)(x + 3)} + \frac{x}{x + 3} = \frac{(x + 2)(x + 3)}{\text{blank}}.
\]

Next, we need to combine the fractions on the left-hand side. To do this, we express the second fraction with a common denominator:

\[
\frac{x}{x + 3} = \frac{x(x + 2)}{(x + 2)(x + 3)}.
\]

Thus, we can now write the left side as:

\[
\frac{5x + x(x + 2)}{(x + 2)(x + 3)} = \frac{5x + x^2 + 2x}{(x + 2)(x + 3)} = \frac{x^2 + 7x}{(x + 2)(x + 3)}.
\]

Now we have:

\[
\frac{x^2 + 7x}{(x + 2)(x + 3)} = \frac{(x + 2)(x + 3)}{\text{blank}}.
\]

For the equality to hold, the left side numerator must equal the right side numerator times some factor that results in the required equation. We examine:

\[
x^2 + 7x = (x + 2)(x + 3).
\]

We find \((x + 2)(x + 3) = x^2 + 5x + 6\). Comparing both sides, we determine:

\[
x^2 + 7x = k(x^2 + 5x + 6).
\]

We compare coefficients or simply find \(k\).

To get \(x^2 + 7x\), we need \(k=1\) to keep it simple, indicating:

\[
\text{blank} = 1.
\]

In this case, the equation would look like:

\[
\frac{x^2 + 7x}{(x + 2)(x + 3)} = \frac{(x + 2)(x + 3)}{1}.
\]

Hence, the answer to the blank is:

\[
\text{blank} = 1.
\]