To factor the quadratic expression [tex]\( -12x^2 - 14x + 10 \)[/tex] using grouping, we follow these steps:
1. Identify and list the coefficients:
The quadratic expression is in the form [tex]\( ax^2 + bx + c \)[/tex], where:
[tex]\[
a = -12, \quad b = -14, \quad c = 10
\][/tex]
2. Multiply [tex]\(a\)[/tex] and [tex]\(c\)[/tex]:
Multiply the coefficient of [tex]\( x^2 \)[/tex] term (which is [tex]\( -12 \)[/tex]) by the constant term (which is [tex]\( 10 \)[/tex]):
[tex]\[
ac = (-12) \times 10 = -120
\][/tex]
3. Find two numbers that multiply to [tex]\( ac \)[/tex] and add to [tex]\( b \)[/tex]:
We need two numbers that multiply to [tex]\(-120\)[/tex] and add up to [tex]\(-14\)[/tex]. These numbers are [tex]\(6\)[/tex] and [tex]\(-20\)[/tex] because:
[tex]\[
6 \times (-20) = -120 \quad \text{and} \quad 6 + (-20) = -14
\][/tex]
4. Break down the middle term using these two numbers:
Rewrite [tex]\(-14x\)[/tex] as [tex]\( 6x - 20x \)[/tex]:
[tex]\[
-12x^2 - 14x + 10 = -12x^2 + 6x - 20x + 10
\][/tex]
5. Group the terms into two pairs:
Group the expression into two pairs:
[tex]\[
(-12x^2 + 6x) + (-20x + 10)
\][/tex]
6. Factor out the greatest common factor (GCF) from each pair:
From the first group [tex]\((-12x^2 + 6x)\)[/tex], factor out [tex]\(-6x\)[/tex]:
[tex]\[
-6x(2x - 1)
\][/tex]
From the second group [tex]\((-20x + 10)\)[/tex], factor out [tex]\(-10\)[/tex]:
[tex]\[
-10(2x - 1)
\][/tex]
7. Factor out the common binomial factor:
Both terms now contain the common factor [tex]\((2x - 1)\)[/tex]:
[tex]\[
-6x(2x - 1) - 10(2x - 1)
\][/tex]
Factor this common binomial factor:
[tex]\[
(-6x - 10)(2x - 1)
\][/tex]
8. Simplify the expression:
Notice we can factor out a common factor of [tex]\(-2\)[/tex] from the first binomial [tex]\((-6x - 10)\)[/tex]:
[tex]\[
-2(3x + 5)(2x - 1)
\][/tex]
Thus, the factored form of the quadratic expression [tex]\( -12x^2 - 14x + 10 \)[/tex] is:
[tex]\[
-2(3x + 5)(2x - 1)
\][/tex]
That is the answer in the form [tex]\( (ax + b)(cx + d) \)[/tex].
