Let P = 12xy - 10y² - 18x², Q = 14x² + 12y² + 9xy, and R = 5y² - x² + xy then…

To add or subtract polynomial expressions, combine only the like terms - terms with the exact same variable part (the same letters raised to the same powers, such as x² with x², y² with y², and xy with xy). Add or subtract just the numerical coefficients of matching terms while the variable part stays unchanged; when subtracting an entire expression, first change the sign of every one of its terms.

Apply this rule to P, Q, and R given in the question:

1. Combine P and Q by adding like terms: the xy terms give 12xy + 9xy = 21xy; the y² terms give -10y² + 12y² = 2y²; the x² terms give -18x² + 14x² = -4x². So P + Q = -4x² + 2y² + 21xy.

2. To subtract R, change the sign of every term of R: -(5y² - x² + xy) = -5y² + x² - xy.

3. Add this to (P + Q), combining like terms again: the x² terms give -4x² + x² = -3x²; the y² terms give 2y² - 5y² = -3y²; the xy terms give 21xy - xy = 20xy.

4. So (P + Q) - R = 20xy - 3x² - 3y².

Cross-check by substituting x = 1, y = 1 into the original expressions: P = 12(1)(1) - 10(1)² - 18(1)² = 12 - 10 - 18 = -16; Q = 14(1)² + 12(1)² + 9(1)(1) = 14 + 12 + 9 = 35; R = 5(1)² - (1)² + (1)(1) = 5 - 1 + 1 = 5. So (P + Q) - R = (-16 + 35) - 5 = 14. Evaluating 20xy - 3x² - 3y² at x = 1, y = 1 gives 20 - 3 - 3 = 14, which matches, confirming the result.

A common slip is not distributing the negative sign to every term of R when subtracting it - for instance leaving R's -x² term unchanged instead of flipping it to +x², or leaving R's +xy term unchanged instead of flipping it to -xy - which changes the sign or value of one or more terms.