Question

1. $81 + 192a^3$

Explanation:

Step1: Identify common factor

Find GCD of 81 and 192.
$\gcd(81,192) = 3$

Step2: Factor out the common term

Extract 3 from each term.
$81 + 192a^3 = 3(27 + 64a^3)$

Step3: Recognize sum of cubes

Rewrite $27+64a^3$ as $3^3+(4a)^3$.
Use sum of cubes formula: $x^3+y^3=(x+y)(x^2-xy+y^2)$

Step4: Apply sum of cubes rule

Substitute $x=3$, $y=4a$.
$27+64a^3=(3+4a)(3^2-3\cdot4a+(4a)^2)=(3+4a)(9-12a+16a^2)$

Step5: Combine all factors

Multiply by the initial common factor.
$3(3+4a)(9-12a+16a^2)$

Answer:

$3(4a + 3)(16a^2 - 12a + 9)$