Questions: Completely factor the expression by grouping, if possible. 10 y^3+45 y^2-2 y-9

Solution Steps

To factor the expression by grouping, we first split the expression into two groups. We then factor out the greatest common factor from each group. If the resulting binomials are the same, we can factor them out, leading to the completely factored expression.

The given expression is \(10y^3 + 45y^2 - 2y - 9\). We can group the terms as follows:

• Group 1: \(10y^3 + 45y^2\)
• Group 2: \(-2y - 9\)

Factor out the greatest common factor from each group:

• From Group 1: \(10y^3 + 45y^2 = 5y^2(2y + 9)\)
• From Group 2: \(-2y - 9 = -1(2y + 9)\)

Since both groups contain the common factor \((2y + 9)\), we can factor it out: \[ 5y^2(2y + 9) - 1(2y + 9) = (2y + 9)(5y^2 - 1) \]

Final Answer