Simplify the expression $\frac{a^2 b^4 - b^2 a^4}{ab(a + b)}$

Let's work through the expression carefully.

Step 1: Factor the numerator

a²b⁴ − b²a⁴ = a²b²(b² − a²)

Here we took out the common factor a²b², since a²b⁴ = a²b²·b² and a⁴b² = a²b²·a².

Step 2: Factor the difference of two squares

b² − a² = (b − a)(b + a)

So the numerator becomes a²b²(b − a)(b + a).

Step 3: Cancel common factors

• a²b² ÷ ab = ab
• (b + a) cancels with (a + b)

This leaves:

ab(b − a) = ab² − a²b

Step 4: Match with the options

ab² − a²b is option D.

Check with numbers: let a = 1, b = 2.

• Numerator: 1·16 − 4·1 = 12
• Denominator: 1·2·3 = 6
• Result: 2
• Option D: 1·4 − 1·2 = 2 ✓

Tip: Whenever you see a difference of powers, first pull out the common monomial factor, then look for a difference of two squares. Watch the order: (b² − a²) is the negative of (a² − b²), which is where students often slip into choosing the wrong sign.