Questions: Factor the following expression. 5 x^2 + 7 x + 2 ([?] x + [])(x + [])

Transcript text: Factor the following expression. \[ \begin{array}{c} 5 x^{2}+7 x+2 \\ ([?] x+[])(x+[]) \end{array} \]

Solution

Solution Steps

To factor the quadratic expression \(5x^2 + 7x + 2\), we need to find two numbers that multiply to the product of the leading coefficient (5) and the constant term (2), which is 10, and add up to the middle coefficient (7). Once these numbers are identified, we can use them to split the middle term and factor by grouping.

Step 1: Identify the Expression

We are given the quadratic expression \(5x^2 + 7x + 2\) to factor.

Step 2: Find Two Numbers

To factor the expression, we need two numbers that multiply to \(5 \times 2 = 10\) and add up to \(7\). These numbers are \(5\) and \(2\).

Step 3: Split the Middle Term

Using the numbers found, we split the middle term \(7x\) into \(5x + 2x\). This gives us: \[ 5x^2 + 5x + 2x + 2 \]

Step 4: Factor by Grouping

Group the terms to factor by grouping: \[ (5x^2 + 5x) + (2x + 2) \]

Factor out the greatest common factor from each group: \[ 5x(x + 1) + 2(x + 1) \]

Step 5: Factor Out the Common Binomial

Notice that \((x + 1)\) is a common factor: \[ (x + 1)(5x + 2) \]