Find the number of ordered pairs $(a,b)$ of integers such that
\[\frac{a + 2}{a + 5} = \frac{b}{4}.\]

We are given the equation

\[
\frac{a + 2}{a + 5} = \frac{b}{4}
\]

and need to find the number of ordered pairs $(a, b)$ of integers that satisfy this equation.

### Step 1: Cross-Multiply to Clear the Fractions
Cross-multiplying gives:

\[
4(a + 2) = b(a + 5)
\]

Expanding both sides:

\[
4a + 8 = ab + 5b
\]

We can rearrange this equation as:

\[
ab + 5b - 4a = 8
\]

### Step 2: Factor the Equation
We can factor the equation as follows:

\[
b(a + 5) = 4a + 8
\]

So:

\[
b = \frac{4a + 8}{a + 5}
\]

### Step 3: Simplify the Expression for $ b $
We simplify the expression for $ b $ by dividing the numerator by the denominator:

\[
b = \frac{4(a + 2)}{a + 5} = \frac{4(a + 2)}{a + 5}
\]

This can be rewritten as:

\[
b = 4 \cdot \frac{a + 2}{a + 5}
\]

Simplify using polynomial long division, or by recognizing the form:

\[
b = 4 \left( 1 - \frac{3}{a+5} \right) = 4 - \frac{12}{a+5}
\]

### Step 4: Ensure $ b $ is an Integer
For $ b $ to be an integer, $ \frac{12}{a+5} $ must be an integer. Thus, $ a + 5 $ must be a divisor of 12.

The divisors of 12 are $ \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 $.

### Step 5: Determine Corresponding $ a $ and $ b $ Values
For each divisor $ d $ of 12, $ a + 5 = d $, so $ a = d - 5 $. Let's compute $ a $ and the corresponding $ b $:

- $ d = 1 $: $ a = -4 $, $ b = 4 - 12/1 = -8 $
- $ d = -1 $: $ a = -6 $, $ b = 4 - 12/(-1) = 16 $
- $ d = 2 $: $ a = -3 $, $ b = 4 - 12/2 = -2 $
- $ d = -2 $: $ a = -7 $, $ b = 4 - 12/(-2) = 10 $
- $ d = 3 $: $ a = -2 $, $ b = 4 - 12/3 = 0 $
- $ d = -3 $: $ a = -8 $, $ b = 4 - 12/(-3) = 8 $
- $ d = 4 $: $ a = -1 $, $ b = 4 - 12/4 = 1 $
- $ d = -4 $: $ a = -9 $, $ b = 4 - 12/(-4) = 7 $
- $ d = 6 $: $ a = 1 $, $ b = 4 - 12/6 = 2 $
- $ d = -6 $: $ a = -11 $, $ b = 4 - 12/(-6) = 6 $
- $ d = 12 $: $ a = 7 $, $ b = 4 - 12/12 = 3 $
- $ d = -12 $: $ a = -17 $, $ b = 4 - 12/(-12) = 5 $

### Step 6: Count the Ordered Pairs
There are 12 different divisors of 12, leading to 12 distinct values of $ a $ and $ b $. Therefore, the number of ordered pairs $(a, b)$ that satisfy the equation is:

\[
\boxed{12}
\]

Would you like further details or have any questions? Here are 8 related questions you might find useful:

1. How do you find the divisors of a number?
2. What is polynomial long division?
3. Can you explain more about solving linear Diophantine equations?
4. How does cross-multiplication work in equations involving fractions?
5. What is the importance of ensuring an expression is an integer in number theory?
6. How do you determine the number of solutions to an equation involving integer solutions?
7. What are the applications of factoring in algebra?
8. How can we visualize the solution set of this equation?

**Tip:** Always check if the variables in your equation are integers, as this often simplifies the problem or constrains the solution set.

Explore how to find integer solutions to the equation (a + 2)/(a + 5) = b/4. This problem involves cross-multiplication and finding integer values for a and b. Suitable for high school students studying equations and fractions.

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