1. **Problem 1: Simplify the polynomial expression** Given: $$P(x) = (3x - 2)(2x^2 + 2x + 3) + 7$$ 2. **Use the distributive property (FOIL for polynomials)** Multiply each term in the first polynomial by each term in the second polynomial: $$(3x)(2x^2) = 6x^3$$ $$(3x)(2x) = 6x^2$$ $$(3x)(3) = 9x$$ $$(-2)(2x^2) = -4x^2$$ $$(-2)(2x) = -4x$$ $$(-2)(3) = -6$$ 3. **Combine all terms:** $$6x^3 + 6x^2 + 9x - 4x^2 - 4x - 6 + 7$$ 4. **Simplify by combining like terms:** $$6x^3 + (6x^2 - 4x^2) + (9x - 4x) + (-6 + 7)$$ $$= 6x^3 + 2x^2 + 5x + 1$$ --- 5. **Problem 2a: Sketch a polynomial function with degree 3, negative leading coefficient, and 2 x-intercepts** - Degree 3 means the highest power of $x$ is 3. - Negative leading coefficient means the $x^3$ term has a negative coefficient. - 2 x-intercepts means the polynomial crosses the x-axis at exactly two points. Example function: $$y = -x^3 + 3x^2 - 2x$$ --- 6. **Problem 2b: Sketch a polynomial function with degree 4, positive leading coefficient, and 3 x-intercepts** - Degree 4 means the highest power of $x$ is 4. - Positive leading coefficient means the $x^4$ term has a positive coefficient. - 3 x-intercepts means the polynomial crosses the x-axis at exactly three points. Example function: $$y = x^4 - 5x^3 + 4x^2$$ --- **Final answers:** $$P(x) = 6x^3 + 2x^2 + 5x + 1$$ Example for 2a: $$y = -x^3 + 3x^2 - 2x$$ Example for 2b: $$y = x^4 - 5x^3 + 4x^2$$