1. **State the problem:** Factor the expression $$7a^2 - 14ab + 7b^2 - 9ac + 9bc + 2c^2$$. 2. **Identify the structure:** Group terms to factor by grouping: $$ (7a^2 - 14ab + 7b^2) + (-9ac + 9bc + 2c^2) $$ 3. **Factor the first group:** $$7a^2 - 14ab + 7b^2 = 7(a^2 - 2ab + b^2) = 7(a - b)^2$$ 4. **Factor the second group:** Rewrite as: $$-9ac + 9bc + 2c^2 = c(-9a + 9b) + 2c^2 = c(9b - 9a) + 2c^2 = c imes 9(b - a) + 2c^2$$ Note that $$9(b - a) = -9(a - b)$$, so: $$c imes 9(b - a) = -9c(a - b)$$ 5. **Rewrite the entire expression:** $$7(a - b)^2 - 9c(a - b) + 2c^2$$ 6. **Let $$x = (a - b)$$ and $$y = c$$, then the expression becomes:** $$7x^2 - 9xy + 2y^2$$ 7. **Factor the quadratic in terms of $$x$$ and $$y$$:** Find two numbers that multiply to $$7 imes 2 = 14$$ and add to $$-9$$: these are $$-7$$ and $$-2$$. Rewrite middle term: $$7x^2 - 7xy - 2xy + 2y^2$$ Group: $$(7x^2 - 7xy) + (-2xy + 2y^2) = 7x(x - y) - 2y(x - y)$$ Factor out common binomial: $$(x - y)(7x - 2y)$$ 8. **Substitute back:** $$(a - b - c)(7(a - b) - 2c)$$ 9. **Simplify second factor:** $$7(a - b) - 2c = 7a - 7b - 2c$$ **Final factored form:** $$ (a - b - c)(7a - 7b - 2c) $$