Let's solve the equation [tex]\(4x^2 + 3x = 24 - x\)[/tex] step by step.
1. First, bring all terms to one side to set the equation to zero:
[tex]\[
4x^2 + 3x - 24 + x = 0
\][/tex]
Combine like terms:
[tex]\[
4x^2 + 4x - 24 = 0
\][/tex]
2. Simplify the equation by dividing all terms by 4:
[tex]\[
x^2 + x - 6 = 0
\][/tex]
3. Factor the quadratic equation [tex]\(x^2 + x - 6\)[/tex]:
We need to find two numbers that multiply to [tex]\(-6\)[/tex] and add to [tex]\(1\)[/tex]. These numbers are [tex]\(3\)[/tex] and [tex]\(-2\)[/tex]:
[tex]\[
(x + 3)(x - 2) = 0
\][/tex]
4. Set each factor equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[
x + 3 = 0 \quad \text{or} \quad x - 2 = 0
\][/tex]
Solving each equation:
[tex]\[
x = -3 \quad \text{or} \quad x = 2
\][/tex]
Therefore, the solutions to the equation [tex]\(4x^2 + 3x = 24 - x\)[/tex] are [tex]\(\boxed{-3\text{ or }2}\)[/tex].
