Question

Factor to solve the equation 4x2+12x+5=−4.(1 point)
Responses

x=112
1 Start Fraction 1 over 2 End Fraction

x=23
Start Fraction 2 over 3 End Fraction

x=−112
negative 1 Start Fraction 1 over 2 End Fraction

x=−23

Answer 1

To solve the equation \( 4x^2 + 12x + 5 = -4 \), we first move the -4 to the left side:

\[
4x^2 + 12x + 5 + 4 = 0
\]

This simplifies to:

\[
4x^2 + 12x + 9 = 0
\]

Next, we can factor the quadratic equation. We look for two numbers that multiply to \( 4 \cdot 9 = 36 \) and add up to \( 12 \). The numbers 6 and 6 fit this requirement.

Thus, we can write the equation as:

\[
4x^2 + 6x + 6x + 9 = 0
\]

Now, we group the terms:

\[
(4x^2 + 6x) + (6x + 9) = 0
\]

Factoring by grouping gives:

\[
2x(2x + 3) + 3(2x + 3) = 0
\]

Now factor out the common term \( (2x + 3) \):

\[
(2x + 3)(2x + 3) = 0
\]
or simplifying further,
\[
(2x + 3)^2 = 0
\]

Now we can set the factor equal to zero:

\[
2x + 3 = 0
\]

Solving for \( x \):

\[
2x = -3
\]
\[
x = -\frac{3}{2}
\]

Thus, the solution to the equation is:

\[
x = -\frac{3}{2}
\]

In terms of your options, that is equivalent to:

\[
x = -2 \frac{1}{2} \quad \text{or} \quad x = -\frac{3}{2}
\]

So the answer is \( x = -2 \frac{1}{2} \) or negative \( \frac{3}{2} \).