Solve -4y^2-16y-12

$$-4 { y }^{ 2 } -16y-12$$

Factor

$$-4\left(y+1\right)\left(y+3\right)$$

Solution Steps

Factor out 4.

$$4\left(-y^{2}-4y-3\right)$$

Consider -y^{2}-4y-3. Factor the expression by grouping. First, the expression needs to be rewritten as -y^{2}+ay+by-3. To find a and b, set up a system to be solved.

$$a+b=-4 ab=-\left(-3\right)=3$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-1 b=-3$$

Rewrite -y^{2}-4y-3 as \left(-y^{2}-y\right)+\left(-3y-3\right).

$$\left(-y^{2}-y\right)+\left(-3y-3\right)$$

Factor out y in the first and 3 in the second group.

$$y\left(-y-1\right)+3\left(-y-1\right)$$

Factor out common term -y-1 by using distributive property.

$$\left(-y-1\right)\left(y+3\right)$$

Rewrite the complete factored expression.

$$4\left(-y-1\right)\left(y+3\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-4y^{2}-16y-12=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$y=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\left(-4\right)\left(-12\right)}}{2\left(-4\right)}$$

Square -16.

$$y=\frac{-\left(-16\right)±\sqrt{256-4\left(-4\right)\left(-12\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$y=\frac{-\left(-16\right)±\sqrt{256+16\left(-12\right)}}{2\left(-4\right)}$$

Multiply 16 times -12.

$$y=\frac{-\left(-16\right)±\sqrt{256-192}}{2\left(-4\right)}$$

Add 256 to -192.

$$y=\frac{-\left(-16\right)±\sqrt{64}}{2\left(-4\right)}$$

Take the square root of 64.

$$y=\frac{-\left(-16\right)±8}{2\left(-4\right)}$$

The opposite of -16 is 16.

$$y=\frac{16±8}{2\left(-4\right)}$$

Multiply 2 times -4.

$$y=\frac{16±8}{-8}$$

Now solve the equation y=\frac{16±8}{-8} when ± is plus. Add 16 to 8.

$$y=\frac{24}{-8}$$

Divide 24 by -8.

$$y=-3$$

Now solve the equation y=\frac{16±8}{-8} when ± is minus. Subtract 8 from 16.

$$y=\frac{8}{-8}$$

Divide 8 by -8.

$$y=-1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -3 for x_{1} and -1 for x_{2}.

$$-4y^{2}-16y-12=-4\left(y-\left(-3\right)\right)\left(y-\left(-1\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$-4y^{2}-16y-12=-4\left(y+3\right)\left(y+1\right)$$

Evaluate

$$-4\left(y+1\right)\left(y+3\right)$$