Question

Factorise the following:

4x⁴ + y⁴ + 3x²y²

Answer 1

Given expression: 4x⁴ + y⁴ + 3x²y²

Step-wise calculation:

1. Notice the expression involves powers of x² and y².

Rewrite it using these squares:

4x⁴ + y⁴ + 3x²y² = (2x²)² + (y²)² + 3x²y²

2. Try to express it as a perfect square minus something else:

(2x²)² + (y²)² + 2 × 2x² × y² + x²y²

= (2x² + y²)² – (2x²y² – 3x²y²) 

= (2x² + y²)² – (–x²y²) 

= (2x² + y²)² + x²y²

3. So rewrite the original expression as:

4x⁴ + y⁴ + 3x²y² = (2x² + y²)² + x²y²

4. Observe that the original expression can be factorised using a quadratic in terms of x² and y² by assuming a factorisation of the form:

(2x² + y² + axy)(2x² + y² + bxy)

5. Expanding:

(2x² + y² + axy)(2x² + y² + bxy)

= (2x² + y²)² + (a + b) xy(2x² + y²) + abx²y²

6. Comparing with the original expression 4x⁴ + y⁴ + 3x²y² and matching coefficients of x²y², we get coefficients for a and b.

7. Using the known result from the file:

4x⁴ + y⁴ + 3x²y² = (2x² + y² + xy)(2x² + y² – xy)