Question

1. $8x^3 - 52x^2 - 96x$

Explanation:

Step1: Factor out the GCF

First, find the greatest common factor (GCF) of the terms \(8x^{3}\), \(-52x^{2}\), and \(-96x\). The GCF of the coefficients \(8\), \(-52\), and \(-96\) is \(4\), and the GCF of the variable terms is \(x\) (since \(x\) is the lowest power of \(x\) present). So, factor out \(4x\):

$$ 8x^{3}-52x^{2}-96x = 4x(2x^{2}-13x - 24) $$

Step2: Factor the quadratic

Now, factor the quadratic \(2x^{2}-13x - 24\). We need two numbers that multiply to \(2\times(-24)=-48\) and add up to \(-13\). The numbers \(-16\) and \(3\) work because \(-16\times3 = -48\) and \(-16 + 3=-13\). Rewrite the middle term using these numbers:

$$ 2x^{2}-16x + 3x - 24 $$

Group the terms:

$$ (2x^{2}-16x)+(3x - 24) $$

Factor out the GCF from each group:

$$ 2x(x - 8)+3(x - 8) $$

Now, factor out \((x - 8)\):

$$ (2x + 3)(x - 8) $$

Step3: Combine the factors

Putting it all together, the factored form of the original polynomial is:

$$ 4x(2x + 3)(x - 8) $$

Answer:

\(4x(2x + 3)(x - 8)\)