1. **State the problem:** Simplify and analyze the expression $a^2 - ab + 2b - 4$ and the expression $a^2 + 2a + 2b - b^2$. 2. **First expression:** $a^2 - ab + 2b - 4$ 3. **Second expression:** $a^2 + 2a + 2b - b^2$ 4. **Note:** These are two separate expressions, not an equation to solve. 5. **For the first expression,** we can try to factor or rearrange terms: $$a^2 - ab + 2b - 4 = a^2 - ab + 2b - 4$$ No obvious common factors or simple factorization. 6. **For the second expression,** rearrange terms: $$a^2 - b^2 + 2a + 2b$$ Recognize $a^2 - b^2$ as a difference of squares: $$a^2 - b^2 = (a - b)(a + b)$$ So, $$a^2 + 2a + 2b - b^2 = (a - b)(a + b) + 2a + 2b$$ 7. **Try to factor further:** Group terms: $$(a - b)(a + b) + 2(a + b) = (a + b)(a - b + 2)$$ 8. **Final factored form:** $$a^2 + 2a + 2b - b^2 = (a + b)(a - b + 2)$$ 9. **Summary:** - First expression $a^2 - ab + 2b - 4$ does not factor nicely with simple methods. - Second expression factors as $(a + b)(a - b + 2)$. 10. **Graph note:** The description suggests a curve crossing the x-axis between 1 and 2, with gentle upward curvature, consistent with a quadratic or similar polynomial.