Factor by grouping. \( 12 y^{2}+17 y+6 \)

To factor the quadratic expression \( 12y^{2} + 17y + 6 \) by grouping, follow these steps: ### Step 1: Multiply the Leading Coefficient and the Constant Term Multiply the coefficient of \( y^2 \) (which is 12) by the constant term (which is 6): \[ 12 \times 6 = 72 \] ### Step 2: Find Two Numbers That Multiply to 72 and Add to 17 We need to find two numbers that multiply to 72 and add up to 17. These numbers are **8** and **9** because: \[ 8 \times 9 = 72 \] \[ 8 + 9 = 17 \] ### Step 3: Rewrite the Middle Term Using These Numbers Express the middle term \( 17y \) as the sum of \( 8y \) and \( 9y \): \[ 12y^{2} + 8y + 9y + 6 \] ### Step 4: Group the Terms Group the terms into two pairs: \[ (12y^{2} + 8y) + (9y + 6) \] ### Step 5: Factor Out the Greatest Common Factor (GCF) from Each Group Factor out the GCF from each pair: - From the first group \( (12y^{2} + 8y) \), the GCF is \( 4y \): \[ 4y(3y + 2) \] - From the second group \( (9y + 6) \), the GCF is \( 3 \): \[ 3(3y + 2) \] ### Step 6: Factor Out the Common Binomial Factor Both groups contain the common binomial factor \( (3y + 2) \): \[ 4y(3y + 2) + 3(3y + 2) = (4y + 3)(3y + 2) \] ### Final Factored Form The expression factored by grouping is: \[ 12y^{2} + 17y + 6 = (4y + 3)(3y + 2) \]