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Chapter 4: Problem 60

Find each product. $$ (4 m+3)(4 m+3) $$

Short Answer

Expert verified
\( 16m^2 + 24m + 9 \)

Step by step solution

01

Write the Expression in Expanded Form

To find the product of \( (4m+3)(4m+3) \), rewrite it as \( (4m+3)^2 \).
02

Use the FOIL Method

Apply the FOIL (First, Outer, Inner, Last) method to expand the expression \( (4m+3)(4m+3) \). \ The First terms are: \( 4m \times 4m = 16m^2 \). \ The Outer terms are: \( 4m \times 3 = 12m \). \ The Inner terms are: \( 3 \times 4m = 12m \). \ The Last terms are: \( 3 \times 3 = 9 \).
03

Combine Like Terms

Add up all the terms obtained from the FOIL method: \( 16m^2 + 12m + 12m + 9 \). \ Combine the like terms to get the final expanded form: \( 16m^2 + 24m + 9 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

FOIL Method
The FOIL method is a handy tool for multiplying two binomials. FOIL stands for First, Outer, Inner, Last. Here鈥檚 how it works with the example \( (4m+3)(4m+3) \):
  • **First:** Multiply the first terms of each binomial: \( 4m \times 4m = 16m^2 \).
  • **Outer:** Multiply the outer terms: \( 4m \times 3 = 12m \).
  • **Inner:** Multiply the inner terms: \( 3 \times 4m = 12m \).
  • **Last:** Multiply the last terms: \( 3 \times 3 = 9 \).
Once you have these products, the next step involves adding them together.
Expansion of Binomials
Expanding binomials means writing out the expression in its full form. In our example, \( (4m + 3)(4m + 3) \) can be written as \( (4m + 3)^2 \). This indicates that we are multiplying \( 4m + 3 \) by itself.

To expand this, you need to:
  • Use the FOIL method to find each product.
  • Combine the terms obtained from these products.
This process results in the expanded binomial: \( 16m^2 + 24m + 9 \).
Combining Like Terms
After applying the FOIL method and expanding your binomial, you end up with several terms. Combining like terms helps to simplify this expression.

In our example, after FOIL, we get: \( 16m^2, 12m, 12m, \) and \( 9 \). The terms \( 12m \) and \( 12m \) are like terms because they both contain \( m \).
  • Add these together: \( 12m + 12m = 24m \).
Finally, combine all the terms: \( 16m^2 + 24m + 9 \). This results in the final expanded form of the binomial.

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