Simplify the Expression: 6x×3y + 4x² - 10y² + 2x(-ay)

To solve this problem, we need to simplify the algebraic expression by performing the multiplications and then combining like terms.

Given expression:
$6x \times 3y + 4x \times x - 5y \times 2y + 2x(-9y)$

Step 1: Simplify each term by performing the multiplications

• First term: $6x \times 3y = (6 \times 3)(x \times y) = 18xy$
• Second term: $4x \times x = 4x^2$
• Third term: $5y \times 2y = (5 \times 2)(y \times y) = 10y^2$
• Fourth term: $2x(-9y) = (2 \times -9)(x \times y) = -18xy$

Step 2: Rewrite the expression with simplified terms
$18xy + 4x^2 - 10y^2 + (-18xy)$
which becomes:
$18xy + 4x^2 - 10y^2 - 18xy$

Step 3: Identify and combine like terms
Like terms are terms that have the same variable parts raised to the same powers.

• Terms with $xy$: $18xy - 18xy = 0$
• Terms with $x^2$: $4x^2$
• Terms with $y^2$: $-10y^2$

Step 4: Write the final simplified expression
After combining like terms, we get:
$4x^2 - 10y^2$

Therefore, the simplified expression is $4x^2 - 10y^2$, which corresponds to choice 4.