Question

1. $7x^2y^6 + 28x^2y^3$
2. $60xy^4 - 54x^3y^5 - 42x^4y^3$
3. $72x^9y^2 - 27x^7y^2 - 18x^7y$
4. $-25 - 45xy^5 + 45x^2$
5. $14xy^7 - 7x^2y^2 - 7x$
6. $-20yx^3 - 100yx^2 + 30y^2$

Explanation:

Step1: Factor out GCF of \(7x^2y^6 + 28x^2y^3\)

Find GCF of coefficients (7,28) is 7. GCF of \(x\) terms: \(x^2\) (lowest power). GCF of \(y\) terms: \(y^3\) (lowest power). So GCF is \(7x^2y^3\).
Divide each term by \(7x^2y^3\): \(\frac{7x^2y^6}{7x^2y^3} + \frac{28x^2y^3}{7x^2y^3} = y^3 + 4\).
Thus, \(7x^2y^6 + 28x^2y^3 = 7x^2y^3(y^3 + 4)\).

Step2: Factor out GCF of \(60xy^4 - 54x^3y^5 - 42x^4y^3\)

GCF of coefficients (60,54,42): GCD(60,54)=6, GCD(6,42)=6. GCF of \(x\) terms: \(x\) (lowest power). GCF of \(y\) terms: \(y^3\) (lowest power). GCF is \(6xy^3\).
Divide each term: \(\frac{60xy^4}{6xy^3} - \frac{54x^3y^5}{6xy^3} - \frac{42x^4y^3}{6xy^3} = 10y - 9x^2y^2 - 7x^3\).
So, \(60xy^4 - 54x^3y^5 - 42x^4y^3 = 6xy^3(10y - 9x^2y^2 - 7x^3)\).

Step3: Factor out GCF of \(72x^9y^2 - 27x^7y^2 - 18x^7y\)

GCF of coefficients (72,27,18): GCD(72,27)=9, GCD(9,18)=9. GCF of \(x\) terms: \(x^7\) (lowest power). GCF of \(y\) terms: \(y\) (lowest power, since \(y^2\) and \(y\), take \(y\)). Wait, recheck: \(y^2\) and \(y\), GCF is \(y\)? No, \(72x^9y^2\) has \(y^2\), \( -27x^7y^2\) has \(y^2\), \( -18x^7y\) has \(y\). So GCF for \(y\) is \(y\). Wait, no: GCF of \(y^2, y^2, y\) is \(y\). Coefficients: 72,27,18. GCD(72,27)=9, GCD(9,18)=9. \(x\): lowest power \(x^7\). So GCF is \(9x^7y\).
Divide each term: \(\frac{72x^9y^2}{9x^7y} - \frac{27x^7y^2}{9x^7y} - \frac{18x^7y}{9x^7y} = 8x^2y - 3y - 2\).
Thus, \(72x^9y^2 - 27x^7y^2 - 18x^7y = 9x^7y(8x^2y - 3y - 2)\). Wait, wait, \(72x^9y^2 / 9x^7y = 8x^2y\), \( -27x^7y^2 /9x^7y = -3y\), \( -18x^7y /9x^7y = -2\). Correct.

Step4: Factor out GCF of \(-25 - 45xy^5 + 45x^2\)

Rearrange: \(45x^2 - 45xy^5 - 25\). GCF of coefficients (45,45,25): GCD(45,45)=45, GCD(45,25)=5. So GCF is 5.
Factor out 5: \(5(9x^2 - 9xy^5 - 5)\).

Step5: Factor out GCF of \(14xy^7 - 7x^2y^2 - 7x\)

GCF of coefficients (14,7,7): 7. GCF of \(x\) terms: \(x\) (lowest power). GCF of \(y\) terms: 1 (since \(y^7, y^2, y^0\)). So GCF is \(7x\).
Divide each term: \(\frac{14xy^7}{7x} - \frac{7x^2y^2}{7x} - \frac{7x}{7x} = 2y^7 - xy^2 - 1\).
Thus, \(14xy^7 - 7x^2y^2 - 7x = 7x(2y^7 - xy^2 - 1)\).

Step6: Factor out GCF of \(-20yx^3 - 100yx^2 + 30y^2\)

Rearrange: \(-20x^3y - 100x^2y + 30y^2\). GCF of coefficients (20,100,30): 10, but with negative? Factor out \(-10y\) (or 10y, but let's take -10y for positive leading coefficient in bracket). Wait, GCF of coefficients: GCD(20,100,30)=10. GCF of \(y\) terms: \(y\), \(x\) terms: 1 (lowest power \(x^0\) for last term). So GCF is \(10y\) (or \(-10y\)). Let's take 10y:
\(10y(-2x^3 - 10x^2 + 3y)\). Or factor out -10y: \(-10y(2x^3 + 10x^2 - 3y)\). Either is correct, but 10y is GCF. Divide: \(\frac{-20x^3y}{10y} - \frac{100x^2y}{10y} + \frac{30y^2}{10y} = -2x^3 - 10x^2 + 3y\). So, \(-20yx^3 - 100yx^2 + 30y^2 = 10y(-2x^3 - 10x^2 + 3y)\) or \(-10y(2x^3 + 10x^2 - 3y)\).

Answer:

1. \(7x^2y^3(y^3 + 4)\)
2. \(6xy^3(10y - 9x^2y^2 - 7x^3)\)
3. \(9x^7y(8x^2y - 3y - 2)\)
4. \(5(9x^2 - 9xy^5 - 5)\)
5. \(7x(2y^7 - xy^2 - 1)\)
6. \(10y(-2x^3 - 10x^2 + 3y)\) (or \(-10y(2x^3 + 10x^2 - 3y)\))