Verify if (3y+x)(4+2x) = 2x²+6xy+4x+12y: Polynomial Equality Check

To determine if the equality $(3y+x)(4+2x)=2x^2+6xy+4x+12y$ is correct, we need to expand and simplify the left-hand side to see if it equals the right-hand side.

First, we expand $(3y+x)(4+2x)$ using the distributive property:

• Multiply $3y$ by $4$, giving $12y$.

• Multiply $3y$ by $2x$, giving $6xy$.

• Multiply $x$ by $4$, giving $4x$.

• Multiply $x$ by $2x$, giving $2x^2$.

Combining all these terms, the left-hand side expands to: $12y + 6xy + 4x + 2x^2$.

Notice that this is precisely the same as the right-hand side: $2x^2 + 6xy + 4x + 12y$.

Therefore, the equality $(3y+x)(4+2x) = 2x^2 + 6xy + 4x + 12y$ holds true.

Thus, the solution to the problem is: $(3y+x)(4+2x)=2x^2+6xy+4x+12y$ is correct, and the answer choice is Yes.