1. **State the problem:** Factor the quadratic expression $$49w^2 + 42wv + 9v^2$$. 2. **Formula and rules:** This is a quadratic trinomial in two variables. We look for factors of the form $$(aw + bv)(cw + dv)$$ such that when expanded, it equals the original expression. 3. **Identify coefficients:** - Coefficient of $w^2$ is 49. - Coefficient of $wv$ is 42. - Coefficient of $v^2$ is 9. 4. **Find factors of 49 and 9:** - 49 factors as $7 \times 7$. - 9 factors as $3 \times 3$. 5. **Try factorization:** Assume $$(7w + 3v)(7w + 3v)$$. 6. **Expand to check:** $$ (7w + 3v)(7w + 3v) = 49w^2 + 21wv + 21wv + 9v^2 = 49w^2 + 42wv + 9v^2 $$ 7. **Conclusion:** The expression factors as $$ (7w + 3v)^2 $$. **Final answer:** $$49w^2 + 42wv + 9v^2 = (7w + 3v)^2$$