For the polynomial function \( f(x)=x^{4}+4 x^{3}+4 x^{2} \), answer the parts a through \( e \). A. The \( x \)-intercept(s) at which the graph crosses the \( x \)-axis is/are \( \square \) 1. (Type an inleger or a decimal. Uso a comma lo separate answers as needed. Type each answer only once.) B. There are no \( x \)-intercepts at which the graph crosses the \( x \)-axis. At which \( x \)-intercept(s) does the graph touch the \( x \)-axis and furn around? Select the corract chcice bolow end, it nocossary, fit in the answer box to completo your choice. A. The \( x \)-inlerceptis) at which the graph touches the \( x \)-axis and turns around isfare \( -2,0 \). (Type an inloger or a docimal. Use a comma to separate answers as needed Type each answer conly once.) B. There are no \( x \)-infercepts at which the graph toaches the \( x \)-axis and lurns around. c. Find the \( y \)-intercepl. The y intorcept is 0 (Sumplify your answet. Type an inleger or a decimal.) d. Determine whether the graph has y-axis symmetry, origin symutiry, or neither. Chooso the cortect answor below. A. The graph of \( I \) is symmetric about the nrigin. B. The graph of \( I \) is symmetric about the \( y \) axis. C. The graph of \( I \) is neither symmerric aboul the \( y \)-axis nor symmetric aboul the origin.

1. Factor the function: \[ f(x)=x^4+4x^3+4x^2=x^2(x^2+4x+4)=x^2(x+2)^2. \] 2. Identify the zeros: - \(x^2=0\) gives \(x=0\) (multiplicity 2). - \((x+2)^2=0\) gives \(x=-2\) (multiplicity 2). 3. Since both zeros have even multiplicities, at both \(x=-2\) and \(x=0\) the graph touches the \(x\)-axis and turns around rather than crossing it. 4. Answers: - **Part A:** Determine where the graph crosses the \(x\)-axis. Since neither \(x=0\) nor \(x=-2\) produces a crossing (each is a touch-point), the correct answer is: **B. There are no \(x\)-intercepts at which the graph crosses the \(x\)-axis.** - **Part B:** Determine where the graph touches and turns around at the \(x\)-axis. The graph touches at both \(x=-2\) and \(x=0\), so the correct answer is: **A. \(-2, 0\)** - **Part C:** Find the \(y\)-intercept. Evaluate \(f(0)\): \[ f(0)=0^4+4(0)^3+4(0)^2=0. \] The \(y\)-intercept is **0**. - **Part D:** Determine symmetry. Check for \(y\)-axis symmetry by testing if \(f(-x)=f(x)\): \[ f(-x)=(-x)^4+4(-x)^3+4(-x)^2=x^4-4x^3+4x^2 \neq f(x). \] Check for origin symmetry by testing if \(f(-x)=-f(x)\): \[ f(-x)=x^4-4x^3+4x^2 \neq -\left(x^4+4x^3+4x^2\right). \] The graph has **neither** \(y\)-axis symmetry nor origin symmetry, so the correct answer is: **C. The graph of \(f\) is neither symmetric about the \(y\)-axis nor symmetric about the origin.**