e) \( 3 x^{2}+4 x+1 \)

It appears you have provided the quadratic expression: \[ 3x^{2} + 4x + 1 \] Assuming your goal is to **factor** this quadratic, here's how you can do it step-by-step. ### Step 1: Identify Coefficients For a quadratic of the form \( ax^{2} + bx + c \): - \( a = 3 \) - \( b = 4 \) - \( c = 1 \) ### Step 2: Find Two Numbers that Multiply to \( a \times c \) and Add to \( b \) Calculate \( a \times c = 3 \times 1 = 3 \). We need two numbers that: - Multiply to **3** - Add up to **4** The numbers **3** and **1** satisfy these conditions: - \( 3 \times 1 = 3 \) - \( 3 + 1 = 4 \) ### Step 3: Rewrite the Middle Term Using These Numbers Express the middle term \( 4x \) as \( 3x + 1x \): \[ 3x^{2} + 3x + 1x + 1 \] ### Step 4: Factor by Grouping Group the terms: \[ (3x^{2} + 3x) + (1x + 1) \] Factor out the greatest common factor (GCF) from each group: \[ 3x(x + 1) + 1(x + 1) \] ### Step 5: Factor Out the Common Binomial Factor Notice that \( (x + 1) \) is a common factor: \[ (3x + 1)(x + 1) \] ### Final Factored Form Thus, the factored form of the quadratic is: \[ 3x^{2} + 4x + 1 = (3x + 1)(x + 1) \] ### Verification To ensure the factorization is correct, you can expand the factors: \[ \begin{align*} (3x + 1)(x + 1) &= 3x \cdot x + 3x \cdot 1 + 1 \cdot x + 1 \cdot 1 \\ &= 3x^{2} + 3x + x + 1 \\ &= 3x^{2} + 4x + 1 \end{align*} \] The expanded form matches the original quadratic, confirming that the factorization is correct.