Verified Answer :

  • Use the algebraic identity \((a+b)^2 = a^2 + 2ab + b^2\) to expand the expression \((-x-5)^2\).
  • Simplify the terms: \(x^2 + 10x + 25\).
  • Identify the coefficient of the \(x\) term in the expanded polynomial.
  • The coefficient is \(\boxed{10}\).

Explanation

  1. Analyzing the Expression

The expression provided is \((-x - 5)^2\). To find the coefficient of the \(x\) term, we must first expand the binomial expression. A useful algebraic identity for squaring a binomial is \((a + b)^2 = a^2 + 2ab + b^2\). In our case, we can set \(a = -x\) and \(b = -5\).

  1. Expanding the Expression

Applying the identity \((a + b)^2 = a^2 + 2ab + b^2\):
\[(-x - 5)^2 = (-x)^2 + 2(-x)(-5) + (-5)^2\]
Calculating each term:

  • \((-x)^2 = x^2\)
  • \(2(-x)(-5) = 10x\)
  • \((-5)^2 = 25\)

So, the expanded form is \(x^2 + 10x + 25\). Alternatively, we can factor out \(-1\) from the expression: \((-1(x + 5))^2 = (-1)^2(x + 5)^2 = 1(x + 5)^2 = x^2 + 10x + 25\). Both methods yield the same result.

  1. Identifying the Coefficient

After expanding \((-x - 5)^2\), we obtain the polynomial \(x^2 + 10x + 25\). The coefficient of the \(x\) term is the number multiplying \(x\), which is \(10\). Comparing this to the given options (\(-25, -10, 10, 25\)), the correct coefficient is \(10\).

Examples

Expanding binomials is useful in various real-world scenarios, such as calculating the area of a square patch where each side length is represented as a linear function. If a square garden has side lengths defined as \((-x - 5)\) meters (assuming \(x\) results in a positive side length in practice), the area would be represented by the expression \((-x - 5)^2 = x^2 + 10x + 25\) square meters. Understanding how to expand these expressions allows for quick modeling of geometric properties and resource allocation.

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