1. **Problem statement:** Factorise the quadratic expression $$6x^2 + 11xy + 5y^2$$. 2. **Formula and approach:** To factorise a quadratic expression of the form $$ax^2 + bxy + cy^2$$, we look for two binomials $$(mx + ny)(px + qy)$$ such that: - $m \times p = a$ - $n \times q = c$ - $m \times q + n \times p = b$ 3. **Identify coefficients:** Here, $a=6$, $b=11$, and $c=5$. 4. **Find factor pairs:** - Factors of $a=6$ are $(6,1)$ or $(3,2)$. - Factors of $c=5$ are $(5,1)$ or $(1,5)$. 5. **Try combinations to get middle term 11:** - Using $(3x + 5y)(2x + y)$: - Multiply outer and inner terms: $3x \times y = 3xy$, $5y \times 2x = 10xy$. - Sum: $3xy + 10xy = 13xy$ (too large). - Using $(3x + y)(2x + 5y)$: - Outer and inner: $3x \times 5y = 15xy$, $y \times 2x = 2xy$. - Sum: $15xy + 2xy = 17xy$ (too large). - Using $(6x + 5y)(x + y)$: - Outer and inner: $6x \times y = 6xy$, $5y \times x = 5xy$. - Sum: $6xy + 5xy = 11xy$ (correct). 6. **Verify the product:** $$ (6x + 5y)(x + y) = 6x^2 + 6xy + 5xy + 5y^2 = 6x^2 + 11xy + 5y^2 $$ 7. **Final answer:** $$6x^2 + 11xy + 5y^2 = (6x + 5y)(x + y)$$