Question

factor. choose the correct answer.
$4x^2 - 20x - 25$

a. $(2x - 5)^2$

b. $4(x^2 - 5x - 6) - 1$

c. $(2x - 5)(2x - 5)$

d. $(2x + 5)(2x - 5)$

e. $(4x - 25)(x + 1)$

f. prime

Explanation:

Step1: Analyze Option a

Expand \((2x - 5)^2\) using the formula \((a - b)^2=a^2-2ab + b^2\). Here, \(a = 2x\) and \(b = 5\), so \((2x - 5)^2=(2x)^2-2\times(2x)\times5+5^2 = 4x^2-20x + 25\). But the given expression is \(4x^2-20x - 25\), so option a is incorrect.

Step2: Analyze Option b

Simplify \(4(x^2 - 5x - 6)-1\). First, expand \(4(x^2 - 5x - 6)=4x^2-20x-24\), then subtract 1: \(4x^2-20x-24 - 1=4x^2-20x-25\). But this is not a factored form (it's a simplified form of the expression after some operation, not factoring), so option b is incorrect.

Step3: Analyze Option c

\((2x - 5)(2x - 5)=(2x - 5)^2\), and we already saw in step 1 that \((2x - 5)^2=4x^2-20x + 25
eq4x^2-20x - 25\), so option c is incorrect.

Step4: Analyze Option d

Expand \((2x + 5)(2x - 5)\) using the difference of squares formula \((a + b)(a - b)=a^2-b^2\). Here, \(a = 2x\) and \(b = 5\), so \((2x + 5)(2x - 5)=(2x)^2-5^2=4x^2-25
eq4x^2-20x - 25\), so option d is incorrect.

Step5: Analyze Option e

Expand \((4x - 25)(x + 1)\). Using the distributive property: \(4x\times x+4x\times1-25\times x-25\times1=4x^2+4x-25x-25=4x^2-21x-25
eq4x^2-20x - 25\), so option e is incorrect.

Step6: Analyze Option f

A prime polynomial is a polynomial that cannot be factored into non - constant polynomials with integer coefficients. Since we have checked all other options and none of them factor \(4x^2-20x - 25\) correctly, the polynomial \(4x^2-20x - 25\) is prime.

Answer:

f. prime