Question

factor this expression completely. (4x^2 - 100) (10 points) = 2 = x - 5 = 2x + 10 = x + 5 = 4x - 10 = x - 10 = 4x + 10 = x + 10 = 2x - 10 = 4

Explanation:

Step1: Recognize difference of squares

The expression \(4x^2 - 100\) is a difference of squares, since \(4x^2=(2x)^2\) and \(100 = 10^2\). The formula for factoring a difference of squares is \(a^2 - b^2=(a + b)(a - b)\). So, \(4x^2-100=(2x)^2 - 10^2=(2x + 10)(2x - 10)\).

Step2: Factor out common terms

Now, factor out the greatest common factor from each binomial. For \(2x + 10\), the GCF is 2, so \(2x+10 = 2(x + 5)\). For \(2x - 10\), the GCF is 2, so \(2x - 10=2(x - 5)\).

Step3: Combine the factors

Multiply the factored forms together: \((2x + 10)(2x - 10)=2(x + 5)\times2(x - 5)=4(x + 5)(x - 5)\). But if we just factor step - by - step using the difference of squares first and then factoring out the 2s, we can also present it as \(4x^2-100 = 4(x^2 - 25)=4(x + 5)(x - 5)\) (since \(x^2-25\) is also a difference of squares with \(a = x\) and \(b = 5\)).

Answer:

The completely factored form of \(4x^2-100\) is \(4(x + 5)(x - 5)\) (or alternatively, factoring step - by - step as \((2x + 10)(2x - 10)=2(x + 5)\times2(x - 5)=4(x + 5)(x - 5)\)). If we consider the options given (even though the question's option presentation is a bit unclear, the factored form is \(4(x + 5)(x - 5)\) or equivalent factored forms like \(2(x + 5)\times2(x - 5)\) etc.)