1. **State the problem:** Simplify the expression $y(5x - 3)^2 - 4y(3x + 1) + 9xy$. 2. **Expand each squared term:** - Expand $(5x - 3)^2$ using the formula $(a - b)^2 = a^2 - 2ab + b^2$: $$ (5x - 3)^2 = 25x^2 - 30x + 9 $$ 3. **Multiply the expansions by terms outside the brackets:** - Multiply $y$ by the expansion: $$ y(25x^2 - 30x + 9) = 25x^2 y - 30 x y + 9 y $$ - Distribute $-4y$ over $(3x + 1)$: $$ -4y(3x + 1) = -12xy - 4y $$ 4. **Rewrite the expression including all terms:** $$ 25x^2 y - 30 x y + 9 y - 12 x y - 4 y + 9 x y $$ 5. **Combine like terms:** - For $x^2 y$ term: $25 x^2 y$ - For $x y$ terms: $-30 x y - 12 x y + 9 x y = (-30 - 12 + 9) xy = -33 xy$ - For $y$ terms: $9 y - 4 y = 5 y$ 6. **Final simplified expression:** $$ 25 x^2 y - 33 x y + 5 y $$ **Answer:** $$\boxed{25 x^2 y - 33 x y + 5 y}$$