Let
P = 12xy − 10y² − 18x²,
Q = 14x² + 12y² + 9xy, and
R = 5y² − x² + xy
then (P + Q) − R =

The correct option is 20xy − 3x² − 3y².

Given the algebraic expressions:
$P=12xy-10y^2-18x^2$
$Q=14x^2+12y^2+9xy$
$R=5y^2-x^2+xy$

We need to find the value of $(P+Q)-R$.

Step 1: Calculate $P+Q$

$P+Q=(12xy-10y^2-18x^2)+(14x^2+12y^2+9xy)$

Group the like terms together:
$P+Q=(-18x^2+14x^2)+(-10y^2+12y^2)+(12xy+9xy)$

Combine the terms:
$P+Q=-4x^2+2y^2+21xy$

Step 2: Subtract $R$ from $(P+Q)$

$(P+Q)-R=(-4x^2+2y^2+21xy)-(5y^2-x^2+xy)$

Distribute the negative sign:
$(P+Q)-R=-4x^2+2y^2+21xy-5y^2+x^2-xy$

Group the like terms together again:
$(P+Q)-R=(-4x^2+x^2)+(2y^2-5y^2)+(21xy-xy)$

Simplify each component:
$(P+Q)-R=-3x^2-3y^2+20xy$

Rearranging the terms gives:
$20xy-3x^2-3y^2$