Verified Answer :

  • Use the FOIL method to expand the expression \((y-4)(7y+6)\) term by term.
  • Multiply the pairs: First \((y \cdot 7y = 7y^2)\), Outer \((y \cdot 6 = 6y)\), Inner \((-4 \cdot 7y = -28y)\), and Last \((-4 \cdot 6 = -24)\).
  • Combine the like terms: \(6y - 28y = -22y\).
  • Conclude that the expanded form is \(7y^2 - 22y - 24\), which corresponds to option b.
  • Final result: \(\boxed{b}\)

Explanation

  1. Applying the FOIL Method

To multiply the expression \((y-4)(7y+6)\), we can use the distributive property, also known as the FOIL method (First, Outer, Inner, Last), which ensures that each term in the first binomial is multiplied by every term in the second binomial.

  1. First: Multiply the first terms of each binomial: \(y \times 7y = 7y^2\).
  2. Outer: Multiply the outer terms of the product: \(y \times 6 = 6y\).
  3. Inner: Multiply the inner terms of the product: \(-4 \times 7y = -28y\).
  4. Last: Multiply the last terms of each binomial: \(-4 \times 6 = -24\).
  1. Simplifying the Expression

Now, sum the products obtained from each step to form the expanded expression:
\[ 7y^2 + 6y - 28y - 24 \]
Next, identify and combine the like terms, which are \(6y\) and \(-28y\):
\[ 6y - 28y = -22y \]
Substituting this back into the expression, we get:
\[ 7y^2 - 22y - 24 \]

  1. Comparing with Options and Conclusion

Comparing our resulting polynomial \(7y^2 - 22y - 24\) with the given options:

A\(7y^2 + 22y - 24\)
B\(7y^2 - 22y - 24\)
C\(7y^2 - 22y - 10\)
D\(7y^2 - 34y - 24\)

Our result matches option (b). Thus, the correct answer is \(7y^2 - 22y - 24\).

Examples

The distributive property is foundational in algebra and is used extensively in practical scenarios like area calculations. For instance, if you have a rectangular garden plot with a length of \((7y+6)\) meters and a width of \((y-4)\) meters, the area of your garden is the product of its dimensions: \((y-4)(7y+6)\) square meters. By expanding this expression to \((7y^2 - 22y - 24)\) square meters, you can easily calculate the area for any specific value of \(y\). Knowing how to multiply polynomials helps simplify complex algebraic models across various fields, from construction planning to financial analysis.

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