Question

22
Add the rational expressions and simplify your answer.



2
h
+
2
+
5
h
+
5
h+2
2
​
+
h+5
5
​
=

Answer 1

To add the given rational expressions, we must first identify a common denominator and then combine the fractions.

The expression we need to simplify is:

\[
\frac{2}{h} + \frac{2}{h + 5} + \frac{5}{h + 2}
\]

### Step 1: Find the Least Common Denominator (LCD)

The denominators are \( h \), \( h + 5 \), and \( h + 2 \). The least common denominator (LCD) will be:

\[
h(h + 5)(h + 2)
\]

### Step 2: Rewrite each term with the LCD

Now we convert each fraction to have the common denominator:

1. For \( \frac{2}{h} \):

\[
\frac{2}{h} = \frac{2(h + 5)(h + 2)}{h(h + 5)(h + 2)}
\]

2. For \( \frac{2}{h + 5} \):

\[
\frac{2}{h + 5} = \frac{2h(h + 2)}{h(h + 5)(h + 2)}
\]

3. For \( \frac{5}{h + 2} \):

\[
\frac{5}{h + 2} = \frac{5h(h + 5)}{h(h + 5)(h + 2)}
\]

### Step 3: Combine the fractions

Now, we add all three fractions together:

\[
\frac{2(h + 5)(h + 2) + 2h(h + 2) + 5h(h + 5)}{h(h + 5)(h + 2)}
\]

### Step 4: Expand the numerators

1. Expand \( 2(h + 5)(h + 2) \):

\[
2[(h + 5)(h + 2)] = 2[h^2 + 2h + 5h + 10] = 2[h^2 + 7h + 10] = 2h^2 + 14h + 20
\]

2. Expand \( 2h(h + 2) \):

\[
2h(h + 2) = 2h^2 + 4h
\]

3. Expand \( 5h(h + 5) \):

\[
5h(h + 5) = 5h^2 + 25h
\]

### Step 5: Combine all expanded terms in the numerator

Now we sum all the terms:

\[
2h^2 + 14h + 20 + 2h^2 + 4h + 5h^2 + 25h
\]

Combine like terms:

\[
(2h^2 + 2h^2 + 5h^2) + (14h + 4h + 25h) + 20 = (9h^2) + (43h) + 20
\]

Thus, the combined expression is:

\[
\frac{9h^2 + 43h + 20}{h(h + 5)(h + 2)}
\]

### Step 6: Factor the numerator (if possible)

Next, we factor \( 9h^2 + 43h + 20 \). We can look for two numbers that multiply to \( 9 \times 20 = 180 \) and add up to \( 43 \).

After checking, we find that \( 3 \) and \( 60 \) fit:

So, we factor the numerator:

\[
9h^2 + 43h + 20 = (9h + 2)(h + 10)
\]

Finally, substituting back gives us:

\[
\frac{(9h + 2)(h + 10)}{h(h + 5)(h + 2)}
\]

### Final Answer

The simplified expression is:

\[
\frac{(9h + 2)(h + 10)}{h(h + 5)(h + 2)}
\]