Factor: $2x^2 + 2x - 112$

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31-10-2026
Mathematics

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Factor: $2x^2 + 2x - 112$

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-10-31T21:11:17-04:00

First, factor out the greatest common factor $2$ from the expression, giving $2(x^2 + x - 56)$.
Next, find two numbers that multiply to $-56$ and add to $1$, which are $8$ and $-7$.
Write the quadratic as a product of binomials: $(x + 8)(x - 7)$.
Combine the GCF with the binomials for the final result: $\boxed{2(x+8)(x-7)}$.

Explanation

Analyzing the expression

We are tasked with factoring the quadratic expression $2x^2 + 2x - 112$. Factoring a quadratic expression involves breaking it down into a product of simpler polynomials, typically binomials. Our first objective is to identify any common factors present in all terms of the expression. Looking at the coefficients 2, 2, and -112, we can see that each term is divisible by 2.

Factoring out the GCF

We begin by factoring out the greatest common factor (GCF), which is 2, from the entire expression:
\[2x^2 + 2x - 112 = 2(x^2 + x - 56).\]
This simplifies the problem to factoring the quadratic trinomial inside the parentheses, $x^2 + x - 56$.

Finding the factors of the trinomial

To factor the trinomial $x^2 + x - 56$, we look for two numbers, $m$ and $n$, such that their product is the constant term ($-56$) and their sum is the coefficient of the linear term ($1$).
Testing factors of $56$:

$-1 \times 56 = -56$, $-1 + 56 = 55$
$-2 \times 28 = -56$, $-2 + 28 = 26$
$-4 \times 14 = -56$, $-4 + 14 = 10$
$-7 \times 8 = -56$, $-7 + 8 = 1$

The numbers $-7$ and $8$ satisfy both conditions.

Final factoring step

Now that we have found the two numbers, $-7$ and $8$, we can write the trinomial $x^2 + x - 56$ as the product of two binomials:
\[x^2 + x - 56 = (x - 7)(x + 8).\]
Finally, including the GCF we factored out earlier, the fully factored form of the original expression is:
\[2(x - 7)(x + 8).\]

Examples

Factoring quadratic expressions is a fundamental skill in algebra with many real-world applications. For example, if you are designing a rectangular garden and know that the area can be represented by the expression $2x^2 + 2x - 112$, factoring this expression helps you determine the dimensions (length and width) of the garden in terms of $x$. By finding factors like $(x+8)$ and $(x-7)$, you can easily calculate how the area changes as you vary the value of $x$, which is helpful in optimizing space or predicting material costs.