Question

In the following exercises, multiply.

$$\begin{gathered} \left(a\right)-5m\left(m^2+3m-18\right) \\ \left(b\right)-3x^2{y}^2\left(7x^2+10xy-y^2\right) \end{gathered}$$

Short answer

The solution is

$$\begin{gathered} \left(a\right)-5m\left(m^2+3m-18\right)=-5m^3-15m^2+90m \\ \left(b\right)-3x^2{y}^2\left(7x^2+10xy-y^2\right)=-21x^4{y}^2-30x^3{y}^3+3x^2{y}^4 \end{gathered}$$

Step-by-step solution

Step 1. Apply the concept of Distributive property

If $A,BandC$are any real numbers or algebraic expressions then

$A\left(B+C\right)=\left(B+C\right)A=AB+AC$

Part (a) Step 1. Given information

The given expression is

$-5m\left(m^2+3m-18\right)$

Part (a) Step 2. Simplify

Using the Distributive property,

$$\begin{gathered} -5m\left(m^2+3m-18\right) \\ =-5m·m^2-5m·3m+5m·18 \\ =-5m^3-15m^2+90m \end{gathered}$$

Part (b) Step 1. Given expression

The given expression is

$-3x^2{y}^2\left(7x^2+10xy-y^2\right)$

Part (b) Step 2. Simplify

Using Distributive property,

$$\begin{gathered} -3x^2{y}^2\left(7x^2+10xy-y^2\right) \\ =-3x^2{y}^2·7x^2-3x^2{y}^2·10xy+3x^2{y}^2·y^2 \\ =-21x^4{y}^2-30x^3{y}^3+3x^2{y}^4 \end{gathered}$$