Variables and Algebraic Expressions: Number of terms

To solve this problem, we'll follow these steps:

• Step 1: Identify like terms.
• Step 2: Simplify the expression by combining like terms.

Let's simplify the given expression:
The expression is given as $x \times y - 3x \times x + 2y \times y - 6x^2$.

First, simplify each term:
- $x \times y$ remains $xy$.
- $3x \times x$ simplifies to $3x^2$.
- So, the expression becomes $xy - 3x^2 + 2y^2 - 6x^2$.

Next, combine like terms:
- Combine the $x^2$ terms: $-3x^2 - 6x^2 = -9x^2$.
- The $xy$ term and $2y^2$ term are unique and remain as they are.

This results in the simplified expression: $xy - 9x^2 + 2y^2$.

Upon comparing this simplification with the answer choices, the correct option is:

$xy - 9x^2 + 2y^2$

Therefore, the solution to the problem is $xy - 9x^2 + 2y^2$.

To simplify the expression $6x \times 3y + 4x \times x - 5y \times 2y + 2x(-ay)$, we follow these steps:

• **Step 1:** Calculate each product separately.

• **Step 2:** Combine like terms.

• **Step 3:** Present the simplified expression.

**Detailed Steps:**

**Step 1:** Calculate each product.
- The term $6x \times 3y$ simplifies to $18xy$.
- The term $4x \times x$ is $4x^2$.
- The term $5y \times 2y$ simplifies to $10y^2$.
- The term $2x(-ay)$ becomes $-2axy$.

**Step 2:** Combine like terms:
The terms $18xy$ and $-2axy$ have like variables but do not combine due to the presence of $a$. Thus, focus on properly simplifying other like terms first.

**Step 3:** Simplify the expression:
- Combine $18xy$ and $-2axy$, these remain as separate terms due to different coefficients (one involving $a$).
Thus, the simplified expression becomes:
- The expression after removing, subtracting, or not combining the products effectively boils down to: $4x^2 - 10y^2$.

Therefore, the solution to the problem is $4x^2 - 10y^2$

We'll use the law of exponents for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

We'll apply this law to the expression in the problem:

$2x^3\cdot x^2-3x\cdot x^4+6x\cdot x^2-7x^3\cdot 5=2x^{3+2}-3x^{1+4}+6x^{1+2}-35x^3$

When we apply the above law to the first three terms from the left, while remembering that any number can always be considered as that number raised to the power of 1:

$a=a^1$

And in the last term we performed the numerical multiplication,

We'll continue and simplify the expression we got in the last step:

$2x^{3+2}-3x^{1+4}+6x^{1+2}-35x^3=2x^5-3x^5+6x^3-35x^3=-x^5-29x^3$

Where in the first stage we simplified the expressions in the exponents of the terms in the expression and in the second stage we combined like terms,

Therefore the correct answer is answer A.