Find the values of a and b if 9x⁴ + 6x³ + 19x² + ax + b is a perfect square.

1. 1 · Write what is given and what is to be found. Given: 9x⁴ + 6x³ + 19x² + ax + b is a perfect square. To find: a and b.
2. 2 · Write the method. Use the long division method. For a perfect square, the remainder is zero.
3. 3 · Do the long division.
   • Root term 3x²; remainder 6x³ + 19x²
   • Divisor 6x² + x; root term x; remainder 18x² + ax + b
   • Divisor 6x² + 2x + 3; root term 3; remainder (a − 6)x + (b − 9)
4. 4 · Set the remainder to zero. a − 6 = 0 and b − 9 = 0.
5. 5 · Check and box the answer. (3x² + x + 3)² = 9x⁴ + 6x³ + 19x² + 6x + 9. Box: a = 6, b = 9.

Scoring tip: Set out the long division in neat columns, like a number square root. Write the remainder clearly before you equate it to zero.