Identify quadratic expression: Identify the quadratic expression to be factored. The given expression is 2x^2 + x - 45.Determine factors: Determine the factors of the quadratic coefficient (2) and the constant term (-45). The factors of 2 are 1 and 2. The factors of -45 are ±1, ±3, ±5, ±9, ±15, ±45. We need to find a pair of factors that multiply to -90 (2 * -45) and add up to 1 (the coefficient of x).Find correct pair: Find the correct pair of factors. After checking possible combinations, we find that 10 and -9 are the factors that meet the criteria. 10 * -9 = -90 and 10 + (-9) = 1.Write middle term: Write the middle term (x) as the sum of two terms using the factors 10 and -9. 2x^2 + x - 45 becomes 2x^2 + 10x - 9x - 45.Group terms: Group the terms to factor by grouping. (2x^2 + 10x) + (-9x - 45)Factor out common factor: Factor out the greatest common factor from each group. 2x(x + 5) - 9(x + 5)Factor out binomial factor: Factor out the common binomial factor (x + 5). (2x - 9)(x + 5)

Algebra 2

Add, subtract, multiply, and divide polynomials

Full solution

Q. (

10

)

2 x^{2}+x-45

Identify quadratic expression: Identify the quadratic expression to be factored. $\newline$ The given expression is $2x^2 + x - 45$ .
Determine factors: Determine the factors of the quadratic coefficient $2$ and the constant term $-45$ . The factors of $2$ are $1$ and $2$ . The factors of $-45$ are $\pm1$ , $\pm3$ , $\pm5$ , $\pm9$ , $-45$ $0$ , $-45$ $1$ . We need to find a pair of factors that multiply to $-45$ $2$ ( $-45$ $3$ ) and add up to $1$ (the coefficient of $-45$ $5$ ).
Find correct pair: Find the correct pair of factors. $\newline$ After checking possible combinations, we find that $10$ and $-9$ are the factors that meet the criteria. $\newline$ $10 \times -9 = -90$ and $10 + (-9) = 1$ .
Write middle term: Write the middle term $x$ as the sum of two terms using the factors $10$ and $-9$ . $2x^2 + x - 45$ becomes $2x^2 + 10x - 9x - 45$ .
Group terms: Group the terms to factor by grouping. $\newline$ $(2x^2 + 10x) + (-9x - 45)$
Factor out common factor: Factor out the greatest common factor from each group. $2x(x + 5) - 9(x + 5)$
Factor out binomial factor: Factor out the common binomial factor $(x + 5)$ . $\newline$ $(2x - 9)(x + 5)$