Questions: Multiply the polynomials. 5ab(2ab+2a-6b)=

Transcript text: Multiply the polynomials. \[ 5 a b(2 a b+2 a-6 b)= \]

Solution

Solution Steps

To multiply the polynomials, distribute the term \(5ab\) across each term inside the parentheses. This involves multiplying \(5ab\) by each term in the expression \(2ab + 2a - 6b\).

Step 1: Define the Polynomials

We start with the polynomials given in the problem: \[ \text{polynomial1} = 5ab \] \[ \text{polynomial2} = 2ab + 2a - 6b \]

Step 2: Distribute and Multiply

Next, we distribute \(5ab\) across each term in the polynomial \(2ab + 2a - 6b\): \[ 5ab \cdot (2ab) + 5ab \cdot (2a) + 5ab \cdot (-6b) \]

Step 3: Calculate Each Term

Calculating each term gives us: \[ 5ab \cdot 2ab = 10a^2b^2 \] \[ 5ab \cdot 2a = 10a^2b \] \[ 5ab \cdot (-6b) = -30ab^2 \]

Step 4: Combine the Results

Combining all the terms results in: \[ 10a^2b^2 + 10a^2b - 30ab^2 \]

Final Answer

The final expression after multiplying the polynomials is: \[ \boxed{10a^2b^2 + 10a^2b - 30ab^2} \]

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