Question

factor to solve the equation 4x2+12x+5=−4

Answer 1

To solve the equation \( 4x^2 + 12x + 5 = -4 \), we first need to move all terms to one side of the equation:

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

This simplifies to:

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

Now, we can factor the quadratic equation. We are looking for two numbers that multiply to \( 4 \cdot 9 = 36 \) and add up to \( 12 \). The numbers \( 6 \) and \( 6 \) meet these criteria.

Thus, we can rewrite the middle term:

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

Next, we can factor by grouping:

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

Factoring out the common terms in each group:

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

Now we can factor out the common binomial factor \( (2x + 3) \):

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

Or simply:

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

Setting the factor equal to zero gives us:

\[
2x + 3 = 0
\]

Solving for \( x \):

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

Thus, the solution to the equation \( 4x^2 + 12x + 5 = -4 \) is:

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