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

To find the number of ordered pairs $(a, b)$ of integers such that

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

we start by cross-multiplying to eliminate the fractions:

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

Expanding both sides, we get:

\[
4a + 8 = ba + 5b.
\]

Rearrange the equation:

\[
4a + 8 = ba + 5b \implies 4a - ba = 5b - 8.
\]

Factor out $a$ on the left-hand side:

\[
a(4 - b) = 5b - 8.
\]

Solve for $a$:

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

For $a$ to be an integer, $\frac{5b - 8}{4 - b}$ must be an integer, which means that $4 - b$ must divide $5b - 8$.

**Step 1: Analyze the Divisibility Condition**

Let's find out for which integer values of $b$, the expression $\frac{5b - 8}{4 - b}$ is an integer.

To do this, note that $5b - 8$ can be rewritten as:

\[
5b - 8 = -(4-b) + 5b + 4.
\]

Now, if $4-b = k$, then

\[
b = 4-k.
\]

Thus, substituting $b = 4 - k$ in the expression for $a$, we have:

\[
a = \frac{5(4-k) - 8}{k}.
\]

Simplifying:

\[
a = \frac{20 - 5k - 8}{k} = \frac{12 - 5k}{k}.
\]

Therefore, $(12 - 5k)$ must be divisible by $k$. Let's analyze the values of $k$ such that this expression is an integer.

**Step 2: Evaluate Possible Integer Values for $k$**

We need $12 - 5k$ to be divisible by $k$:

\[
12 - 5k = mk \quad \text{for some integer } m.
\]

\[
12 = mk + 5k = (m+5)k.
\]

Hence, $k$ must be a divisor of $12$. The divisors of $12$ are $ \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$.

**Step 3: Calculate Corresponding Values of $(a, b)$**

We will check each divisor $k$ and determine the corresponding $a$ and $b$.

- **For $k = 1$:**
  \[
  b = 4-1 = 3,\quad a = \frac{12 - 5 \times 1}{1} = 7.
  \]
  Ordered pair: $(a, b) = (7, 3)$.

- **For $k = -1$:**
  \[
  b = 4 - (-1) = 5,\quad a = \frac{12 - 5 \times (-1)}{-1} = -17.
  \]
  Ordered pair: $(a, b) = (-17, 5)$.

- **For $k = 2$:**
  \[
  b = 4-2 = 2,\quad a = \frac{12 - 5 \times 2}{2} = 1.
  \]
  Ordered pair: $(a, b) = (1, 2)$.

- **For $k = -2$:**
  \[
  b = 4 - (-2) = 6,\quad a = \frac{12 - 5 \times (-2)}{-2} = -11.
  \]
  Ordered pair: $(a, b) = (-11, 6)$.

- **For $k = 3$:**
  \[
  b = 4-3 = 1,\quad a = \frac{12 - 5 \times 3}{3} = -1.
  \]
  Ordered pair: $(a, b) = (-1, 1)$.

- **For $k = -3$:**
  \[
  b = 4 - (-3) = 7,\quad a = \frac{12 - 5 \times (-3)}{-3} = -9.
  \]
  Ordered pair: $(a, b) = (-9, 7)$.

- **For $k = 4$:**
  \[
  b = 4-4 = 0,\quad a = \frac{12 - 5 \times 4}{4} = -2.
  \]
  Ordered pair: $(a, b) = (-2, 0)$.

- **For $k = -4$:**
  \[
  b = 4 - (-4) = 8,\quad a = \frac{12 - 5 \times (-4)}{-4} = -7.
  \]
  Ordered pair: $(a, b) = (-7, 8)$.

- **For $k = 6$:**
  \[
  b = 4-6 = -2,\quad a = \frac{12 - 5 \times 6}{6} = -3.
  \]
  Ordered pair: $(a, b) = (-3, -2)$.

- **For $k = -6$:**
  \[
  b = 4 - (-6) = 10,\quad a = \frac{12 - 5 \times (-6)}{-6} = -3.
  \]
  Ordered pair: $(a, b) = (-3, 10)$.

- **For $k = 12$:**
  \[
  b = 4-12 = -8,\quad a = \frac{12 - 5 \times 12}{12} = -4.
  \]
  Ordered pair: $(a, b) = (-4, -8)$.

- **For $k = -12$:**
  \[
  b = 4 - (-12) = 16,\quad a = \frac{12 - 5 \times (-12)}{-12} = -2.
  \]
  Ordered pair: $(a, b) = (-2, 16)$.

**Conclusion**

The integer values of $k$ yield 12 possible ordered pairs $(a, b)$. Therefore, the number of ordered pairs $(a, b)$ that satisfy the equation is $\boxed{12}$.

Discover how to find the number of ordered pairs (a, b) of integers that satisfy the equation (a + 2) / (a + 5) = b / 4. Follow a detailed step-by-step solution involving fractions, equations, and divisibility. Suitable for students and enthusiasts interested in integer solutions and mathematical problem-solving.

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