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The image shows the following problem in Spanish:

**Problem:**
"Si \( a - b = 3 \) y \( ab = 2 \), halla el valor de \( a^4 + b^4 \)."

Options:
- A) 160
- B) 161
- C) 162
- D) 163
- E) N.A. (Not Applicable)

Let's solve this step by step:

### Step 1: Solve for \( a^2 + b^2 \)
We have two equations:
1. \( a - b = 3 \)
2. \( ab = 2 \)

Using the identity:
\[
(a - b)^2 = a^2 + b^2 - 2ab
\]

Substitute the values:
\[
(3)^2 = a^2 + b^2 - 2(2)
\]
\[
9 = a^2 + b^2 - 4
\]
\[
a^2 + b^2 = 13
\]

### Step 2: Find \( a^4 + b^4 \)
Using the identity:
\[
a^4 + b^4 = (a^2 + b^2)^2 - 2(a^2b^2)
\]
We already know \( a^2 + b^2 = 13 \), so now calculate:
\[
a^2b^2 = (ab)^2 = (2)^2 = 4
\]

Substitute into the formula:
\[
a^4 + b^4 = (13)^2 - 2(4)
\]
\[
a^4 + b^4 = 169 - 8 = 161
\]

Thus, the answer is \( \boxed{161} \).

Would you like further details or have any questions?

### Related questions:
1. What is the general formula for \( a^4 + b^4 \)?
2. How can you solve for \( a^2 + b^2 \) given other values?
3. What other identities are useful in solving powers of sums?
4. Can this method apply to complex numbers?
5. How would the solution change if \( a + b \) were also given?

**Tip**: Identities involving squares and powers can simplify complex expressions—try using them whenever you have two-variable equations.

Si a - b = 3 y ab = 2, halla el valor de a^4 + b^4.

Step-by-step solution for finding the value of a^4 + b^4 when a - b = 3 and ab = 2. This problem covers algebraic concepts and identities related to powers of polynomials, suitable for high school students.

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