6pq - 3rs - 3ps + 6qr = 6pq - 3rs - 3ps + 6qr

6pq - 3rs - 3ps + 6qr = 6pq + 6qr - 3rs - 3ps

6pq - 3rs - 3ps + 6qr = 6q(p + r) - 3s(r + p)

6pq - 3rs - 3ps + 6qr = 6q(p + r) - 3s(p + r)

6pq - 3rs - 3ps + 6qr = (6q - 3s)(p + r)

6pq - 3rs - 3ps + 6qr = 3(2q - s)(p + r)

6pq - 3rs - 3ps + 6qr = 3(p + r)(2q - s)

(2x + 2y) + (2x - 2y)(x + y) = (2x + 2y)(x - y) + (2x - 2y)(x + y)

2x + 2y = 2(x + y)

2x - 2y = 2(x - y)

(2x + 2y) + (2x - 2y)(x + y) = 2(x + y)(x - y) + 2(x - y)(x + y)

2(x - y)(x + y) = 2(x + y)(x - y)

(2x + 2y) + (2x - 2y)(x + y) = 2(x + y)(x - y) +2(x + y)(x - y)

(2x + 2y) + (2x - 2y)(x + y) = 4(x + y)(x - y)

Since 3 can go into 6 and 9, split -6pqr into -3pqr and -3pqr

p²q² - 6pqr + 9r² = p²q² - 3pqr - 3pqr + 9r²

p²q² - 6pqr + 9r² = pq(pq - 3r) - 3r(pq - 3r)

p²q² - 6pqr + 9r² = (pq - 3r)(pq - 3r)

Since pq - 3r = pq - 3r, the multiplication makes it square

p²q² - 6pqr + 9r² = (pq - 3r)²

The equation above is similar to the general factorization rule of (a - b)²

(a - b)² = a² - 2ab + b²

Thus the first term square then the sign (either + or -) and 2 times the first and second terms + the second term square

Similarly (a + b)² = a² + 2ab + b²

The first term for the equation, p²q² - 6pqr + 9r² is pq and the second term is 3r and the sign is -

5p² + 4pq - 15pr - 12qr = 5p² + 4pq - 15pr - 12qr

5p² + 4pq - 15pr - 12qr = p(5p + 4q) - 3r(5p + 4q)

5p² + 4pq - 15pr - 12qr = (5p + 4q)(p - 3r)

6pq - 3rs - 3ps + 6qr = 6pq - 3rs - 3ps + 6qr

6pq - 3rs - 3ps + 6qr = 6pq + 6qr - 3rs - 3ps

6pq - 3rs - 3ps + 6qr = 6q(p + r) - 3s(r + p)

6pq - 3rs - 3ps + 6qr = 6q(p + r) - 3s(p + r)

6pq - 3rs - 3ps + 6qr = (6q - 3s)(p + r)

6pq - 3rs - 3ps + 6qr = 3(2q - s)(p + r)

6pq - 3rs - 3ps + 6qr = 3(p + r)(2q - s)

(2x + 2y) + (2x - 2y)(x + y) = (2x + 2y)(x - y) + (2x - 2y)(x + y)

2x + 2y = 2(x + y)

2x - 2y = 2(x - y)

(2x + 2y) + (2x - 2y)(x + y) = 2(x + y)(x - y) + 2(x - y)(x + y)

2(x - y)(x + y) = 2(x + y)(x - y)

(2x + 2y) + (2x - 2y)(x + y) = 2(x + y)(x - y) +2(x + y)(x - y)

(2x + 2y) + (2x - 2y)(x + y) = 4(x + y)(x - y)

Since 3 can go into 6 and 9, split -6pqr into -3pqr and -3pqr

p²q² - 6pqr + 9r² = p²q² - 3pqr - 3pqr + 9r²

p²q² - 6pqr + 9r² = pq(pq - 3r) - 3r(pq - 3r)

p²q² - 6pqr + 9r² = (pq - 3r)(pq - 3r)

Since pq - 3r = pq - 3r, the multiplication makes it square

p²q² - 6pqr + 9r² = (pq - 3r)²

The equation above is similar to the general factorization rule of (a - b)²

(a - b)² = a² - 2ab + b²

Thus the first term square then the sign (either + or -) and 2 times the first and second terms + the second term square

Similarly (a + b)² = a² + 2ab + b²

The first term for the equation, p²q² - 6pqr + 9r² is pq and the second term is 3r and the sign is -