Question

Multiply. $$(x+4)(x+2)$$

Short answer

Question: Multiply the two binomials \((x+4)\) and \((x+2)\) and simplify the expression. Answer: \(x^2+6x+8\)

Step-by-step solution

Apply the distributive property once

Distribute the first term in the first binomial, \(x\), to both terms in the second binomial: $$x(x+2)=x^2+2x$$

Apply the distributive property again

Distribute the second term in the first binomial, \(4\), to both terms in the second binomial: $$4(x+2)=4x+8$$

Add the two expressions

Now, add the expressions obtained in step 1 and step 2: $$(x^2+2x)+(4x+8)$$

Combine like terms

Combine the like terms (the \(2x\) and the \(4x\) terms): $$x^2+(2x+4x)+8$$ This results in the final expression: $$x^2+6x+8$$

Key concepts

Distributive Property

Understanding the distributive property is crucial in algebra, especially when you're working on multiplying binomials like \((x+4)(x+2)\). This mathematical rule is a way to simplify complex expressions and it states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. In our exercise, we apply this property twice.

First, we distribute the term \(x\) across the binomial \(x+2\), resulting in \(x^2+2x\). Next, we distribute the number \(4\) across the binomial \(x+2\), producing \(4x+8\). These steps show how the distributive property breaks down a seemingly difficult problem into manageable parts.

Combining Like Terms

Once we've distributed terms in our multiplication problem, our next step is to combine like terms. In algebra, like terms are terms that have the same variable raised to the same power. In the given expression \(x^2+2x+4x+8\), we can see that \(2x\) and \(4x\) are like terms because they both contain the variable \(x\) raised to the first power.

To combine them, we simply add their coefficients (the numbers in front of the variable) together. This results in \(2x + 4x = 6x\), significantly simplifying our expression to \(x^2+6x+8\). The process of combining like terms is essential for simplifying algebraic expressions and solving equations efficiently.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. When multiplying binomials, as in our example \((x+4)(x+2)\), we end up creating a new algebraic expression that represents the area of a rectangle with sides \(x+4\) and \(x+2\).

Through the multiplication process, using the distributive property and combining like terms, we simplify the product of the binomials to form this new expression, which in our case is \(x^2+6x+8\). Algebraic expressions are fundamental in representing real-world situations and solving problems abstractly within mathematical models.