Question

what is the most simplified expression for $4c^{2}d + 3dc - 2(dc^{2} + cd) + 6c^{2}d^{2}$?
\bigcirc $2c^{2}d + dc + 6c^{2}d^{2}$
\bigcirc $2c^{2}d + 5dc + 6c^{2}d^{2}$
\bigcirc $3c^{2}d + 3dc + 6c^{2}d^{2} + cd$
\bigcirc $3c^{2}d + 2dc + 6c^{2}d^{2} - 2cd$

Explanation:

Step1: Distribute the -2

We need to distribute the -2 into the parentheses \( -2(dc^{2}+cd) \). Using the distributive property \( a(b + c)=ab+ac \), we get \( -2 \times dc^{2}-2\times cd=-2dc^{2}-2cd \). So the original expression becomes \( 4c^{2}d + 3dc-2dc^{2}-2cd + 6c^{2}d^{2} \).

Step2: Combine like terms for \( c^{2}d \) terms

The terms with \( c^{2}d \) are \( 4c^{2}d \) and \( - 2dc^{2}\) (note that \( dc^{2}=c^{2}d \)). Combining these, we have \( 4c^{2}d-2c^{2}d=(4 - 2)c^{2}d = 2c^{2}d \).

Step3: Combine like terms for \( cd \) terms

The terms with \( cd \) are \( 3dc \) and \( - 2cd \) (note that \( dc = cd \)). Combining these, we have \( 3cd-2cd=(3 - 2)cd=cd \) or \( dc \).

Step4: Write the simplified expression

Putting it all together, the simplified expression is \( 2c^{2}d+dc + 6c^{2}d^{2} \) (since the \( 6c^{2}d^{2}\) term has no like terms to combine with).

Answer:

\( 2c^{2}d+dc + 6c^{2}d^{2} \) (corresponding to the first option: \( 2c^{2}d+dc+6c^{2}d^{2} \))