Factor completely $4 x ^2-8 x -60$.
We are tasked with factoring the quadratic expression \(4x^2 - 8x - 60\) completely. A complete factorization involves identifying common factors across all terms and then further decomposing the remaining quadratic polynomial into a product of simpler linear binomials. The given expression is a quadratic polynomial of the form \(ax^2 + bx + c\), where \(a=4\), \(b=-8\), and \(c=-60\).
First, we identify the greatest common factor (GCF) of the terms \(4x^2\), \(-8x\), and \(-60\). All these coefficients are divisible by 4. By factoring out the GCF of 4, the expression becomes:
\[4x^2 - 8x - 60 = 4(x^2 - 2x - 15)\]
Now we focus on factoring the quadratic expression inside the parentheses, \(x^2 - 2x - 15\). To factor a quadratic of the form \(x^2 + Bx + C\), we search for two integers whose product is \(C = -15\) and whose sum is \(B = -2\).
Looking at the factors of \(-15\):
The pair \(-5\) and \(3\) satisfies both conditions. Therefore, \(x^2 - 2x - 15 = (x - 5)(x + 3)\).
Combining the constant 4 factored out in the first step with the factors of the trinomial, we obtain the fully factored form:
\[4x^2 - 8x - 60 = 4(x - 5)(x + 3)\]
Comparing this result to the given options, we identify that this matches the expression provided in the second choice. While other expressions can represent the polynomial when expanded, the standard "completely factored" form includes the extraction of the constant greatest common factor. The correct choice is \(4(x - 5)(x + 3)\).
Factoring expressions is essential in fields like physics and engineering, where equations must often be simplified to identify roots or equilibrium points. For instance, in modeling the motion of an object under a quadratic force, factoring the resulting polynomial allows you to quickly solve for the moments when the object is at rest (where the force equals zero). Mastering this skill provides a fundamental tool for solving complex algebraic equations efficiently.