Question

1. $45x^{8}+30x^{6}$

$15x^{6}(3x^{2} + 2)$

1. $-15x^{9}+10x^{3}$
2. $-18x^{5}-24x^{4}$
3. $2x^{10}+18x^{2}$
4. $-12x^{8}+54x^{2}$
5. $12x^{10}+24x^{4}$
6. $-8x^{7}+24x^{5}$
7. $-6x^{9}+18x^{8}$

a. $-5x^{3}(x^{6}-2 + 2x^{6})$
d. $-6x^{2}(3x^{3}+4x^{2})$
h. $15x^{6}(3x^{2}+2)$
i. $8x^{4}(2x - x^{3}+x)$
n. $-x^{2}(-2x^{8}-18)$
o. $6x^{2}(8 - 2x^{6}+1)$
p. $4x^{3}(3x^{7}+6x)$
t. $3x^{8}(-2x + 6)$

Explanation:

Step1: Factor out common factors for each expression on the left - hand side.

For \(45x^{8}+30x^{6}\), the greatest common factor of \(45\) and \(30\) is \(15\), and the lowest power of \(x\) is \(x^{6}\). So \(45x^{8}+30x^{6}=15x^{6}(3x^{2} + 2)\) which matches option H.
For \(-15x^{9}+10x^{3}\), the GCF of \(15\) and \(10\) is \(5\), and the lowest power of \(x\) is \(x^{3}\), so \(-15x^{9}+10x^{3}=- 5x^{3}(3x^{6}-2)\) and there is no match among the given options.
For \(-18x^{5}-24x^{4}\), the GCF of \(18\) and \(24\) is \(6\), and the lowest power of \(x\) is \(x^{4}\), so \(-18x^{5}-24x^{4}=-6x^{4}(3x + 4)\) and there is no match among the given options.
For \(2x^{10}+18x^{2}\), the GCF of \(2\) and \(18\) is \(2\), and the lowest power of \(x\) is \(x^{2}\), so \(2x^{10}+18x^{2}=2x^{2}(x^{8}+9)\) and there is no match among the given options.
For \(-12x^{8}+54x^{2}\), the GCF of \(12\) and \(54\) is \(6\), and the lowest power of \(x\) is \(x^{2}\), so \(-12x^{8}+54x^{2}=-6x^{2}(2x^{6}-9)\) and there is no match among the given options.
For \(12x^{10}+24x^{4}\), the GCF of \(12\) and \(24\) is \(12\), and the lowest power of \(x\) is \(x^{4}\), so \(12x^{10}+24x^{4}=12x^{4}(x^{6}+2)\) and there is no match among the given options.
For \(-8x^{7}+24x^{5}\), the GCF of \(8\) and \(24\) is \(8\), and the lowest power of \(x\) is \(x^{5}\), so \(-8x^{7}+24x^{5}=-8x^{5}(x^{2}-3)\) and there is no match among the given options.
For \(-6x^{9}+18x^{8}\), the GCF of \(6\) and \(18\) is \(6\), and the lowest power of \(x\) is \(x^{8}\), so \(-6x^{9}+18x^{8}=-6x^{8}(x - 3)=3x^{8}(-2x + 6)\) which matches option T.

Answer:

1. H. \(15x^{6}(3x^{2}+2)\)
2. No match
3. No match
4. No match
5. No match
6. No match
7. No match
8. T. \(3x^{8}(-2x + 6)\)