Question

In the following exercises, multiply.

$$\begin{gathered} \left(a\right)-8x\left(x^2+2x-15\right) \\ \left(b\right)5p{q}^3\left(p^2-2pq+6q^2\right) \end{gathered}$$

Short answer

The solution is

$$\begin{gathered} \left(a\right)-8x\left(x^2+2x-15\right)=-8x^3-16x^2+120x \\ \left(b\right)5p{q}^3\left(p^2-2pq+6q^2\right)=5p^3{q}^3-10p^2{q}^4+30p{q}^5 \end{gathered}$$

Step-by-step solution

Step 1. Apply the concept of Distributive Property

If $A,B,C$are any real number or algebraic expressions then

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

Part (a) Step 1. Given expression

The given expression is

$-8x\left(x^2+2x-15\right)$

Part (a) Step 2. Simplify

Using Distributive property ,

$$\begin{gathered} -8x\left(x^2+2x-15\right) \\ =\left(-8x\right)·x^2-16x·x+120x \\ =-8x^3-16x^2+120x \end{gathered}$$

Part (b) Step 1. Given information

The given expression is

$5p{q}^3\left(p^2-2pq+6q^2\right)$

Part (b) Step 2. Simplify

Using Distributive property,

$$\begin{gathered} 5p{q}^3\left(p^2-2pq+6q^2\right) \\ =5p{q}^3·p^2-5p{q}^3·2pq+5p{q}^3·6q^2 \\ =5p^3{q}^3-10p^2{q}^4+30p{q}^5 \end{gathered}$$