Find the product. Simplify your answer: -2w^2(-3w^2 + 9)

Understand the Problem

The question asks for the product of the polynomial expression -2w^2 times (-3w^2 + 9) and requires simplification of the result.

Answer

The product is $6w^4 - 18w^2$.

Steps to Solve

Distributing the Monomial We need to distribute the monomial $-2w^2$ to each term in the polynomial $(-3w^2 + 9)$.

This means we will perform two multiplications:

Multiply $-2w^2$ by $-3w^2$
Multiply $-2w^2$ by $9$

Calculating the First Product For the first multiplication: $$ -2w^2 \cdot -3w^2 = 6w^4 $$ The negative signs cancel out and we multiply the coefficients and add the exponents of $w$.
Calculating the Second Product For the second multiplication: $$ -2w^2 \cdot 9 = -18w^2 $$ We just multiply the coefficients, keeping the $w^2$.
Combining the Results Now, we combine the results of the two products: $$ 6w^4 - 18w^2 $$

The simplified product of the expression is $6w^4 - 18w^2$.

More Information

This expression combines the results from distributing a monomial across a polynomial, a common operation in algebra. The terms $6w^4$ and $-18w^2$ cannot be combined further because they are not like terms.

Tips

Forgetting to distribute to both terms in the polynomial.
Incorrectly handling the signs when multiplying the coefficients.

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