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We are given the equation:

\[
(5 - \sqrt{8})(3 - a\sqrt{2}) = b - 21\sqrt{2}
\]

We need to determine the integer values of \(a\) and \(b\). Let's solve this step by step by expanding the left-hand side and comparing terms.

### Step 1: Expand the left-hand side
Using the distributive property (FOIL method), expand the expression:

\[
(5 - \sqrt{8})(3 - a\sqrt{2}) = 5(3 - a\sqrt{2}) - \sqrt{8}(3 - a\sqrt{2})
\]

Expanding each part:

\[
5(3 - a\sqrt{2}) = 15 - 5a\sqrt{2}
\]
\[
-\sqrt{8}(3 - a\sqrt{2}) = -\sqrt{8} \cdot 3 + \sqrt{8} \cdot a\sqrt{2}
\]

Now, simplify these terms:

\[
\sqrt{8} = 2\sqrt{2}
\]

So the expressions become:

\[
-\sqrt{8} \cdot 3 = -6\sqrt{2}
\]
\[
\sqrt{8} \cdot a\sqrt{2} = 2\sqrt{2} \cdot a\sqrt{2} = 2a \cdot 2 = 4a
\]

Thus, the expanded expression is:

\[
(5 - \sqrt{8})(3 - a\sqrt{2}) = 15 - 5a\sqrt{2} - 6\sqrt{2} + 4a
\]

### Step 2: Combine like terms
Now, group the terms involving \(\sqrt{2}\) and the constant terms:

\[
15 + 4a - (5a + 6)\sqrt{2}
\]

The given equation is equal to:

\[
b - 21\sqrt{2}
\]

### Step 3: Equate the rational and irrational parts
We can now compare the rational and irrational parts on both sides of the equation:

- For the rational part: 
  \[
  15 + 4a = b
  \]
- For the irrational part:
  \[
  -(5a + 6) = -21
  \]

### Step 4: Solve for \(a\) and \(b\)

From the irrational part:

\[
5a + 6 = 21 \quad \Rightarrow \quad 5a = 15 \quad \Rightarrow \quad a = 3
\]

Now substitute \(a = 3\) into the equation for \(b\):

\[
15 + 4(3) = b \quad \Rightarrow \quad 15 + 12 = b \quad \Rightarrow \quad b = 27
\]

### Final Answer:
The values of the integers \(a\) and \(b\) are:

\[
a = 3, \quad b = 27
\]

Would you like more details, or do you have any further questions? Here are some related questions you might find interesting:

1. How can we generalize this method to solve similar equations with square roots?
2. What happens if we replace \( \sqrt{2} \) with another irrational number?
3. Can you solve an equation where both sides contain different irrational terms?
4. How would the solution change if we used complex numbers?
5. What are some other techniques to solve equations involving radicals?

**Tip**: Always simplify square roots like \( \sqrt{8} \) to their simplest form (e.g., \( \sqrt{8} = 2\sqrt{2} \)) before expanding.

Given that (5 - √8)(3 - a√2) = b - 21√2, find the values of the integers a and b.

Learn how to solve the algebraic equation (5 - √8)(3 - a√2) = b - 21√2 by expanding terms and equating rational and irrational parts. Detailed step-by-step explanation provided, suitable for students in Grades 9-12.

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