Let's expand the expression **step by step**: --- ### **Step 1: Expand \( (x^2 - 2x + 1) \) by \( x^3 \)** Multiply each term inside the parentheses by \( x^3 \): \[ x^3(x^2 - 2x + 1) = x^3 \cdot x^2 - x^3 \cdot 2x + x^3 \cdot 1 = x^5 - 2x^4 + x^3 \] **Explanation:** We distribute \( x^3 \) to each term in \( (x^2 - 2x + 1) \). --- ### **Step 2: Expand \( (x^2 - 2x + 1) \) by \( 2 \) and add results** Multiply each term in the parentheses by 2: \[ 2(x^2 - 2x + 1) = 2x^2 - 4x + 2 \] Now, add this to the result from Step 1: \[ x^5 - 2x^4 + x^3 + 2x^2 - 4x + 2 \] **Explanation:** We distribute 2 to each term in \( (x^2 - 2x + 1) \) and combine like terms from both expansions. --- ### **Final Answer:** \[ \boxed{y = x^5 - 2x^4 + x^3 + 2x^2 - 4x + 2} \]

Let's expand the expression step by step: --- ### **Step 1: Distribute \( x^3 \) across \( (x^2 - 2x + 1) \)** Multiply each term inside the parentheses by \( x^3 \): \[ x^3 \times x^2 = x^5 \] \[ x^3 \times (-2x) = -2x^4 \] \[ x^3 \times 1 = x^3 \] Adding these results: \[ x^5 - 2x^4 + x^3 \] **Explanation:** We are applying the distributive property, multiplying each term in \( (x^2 - 2x + 1) \) by \( x^3 \). --- ### **Step 2: Distribute 2 across \( (x^2 - 2x + 1) \) and add to the previous result** Multiply each term in the parentheses by 2: \[ 2 \times x^2 = 2x^2 \] \[ 2 \times (-2x) = -4x \] \[ 2 \times 1 = 2 \] Now, combine this with the result from Step 1: \[ x^5 - 2x^4 + x^3 + 2x^2 - 4x + 2 \] **Explanation:** We distribute 2 over the trinomial and then add these terms to the previous expansion to get the complete expanded form. --- ### **Final Answer:** \[ \boxed{y = x^5 - 2x^4 + x^3 + 2x^2 - 4x + 2} \] This is the fully expanded form of \( y = (x^3 + 2)(x^2 - 2x + 1) \).