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Hannah. added one to her age, then multiplied the result by $4$. She squared that product, then increased the result by $3$, then doubled it. The final result was $1214$ more than $60$ times her age. Find Hannah.'s age, which we can assume is a positive number. I simplified to $32x^2+4x-1179=0$ but I don't really know what to do now. Please help!

Letting Hannah's age be $x$ we obtain that

$$2left( left(4(x+1)right)^2 +3 right)-1214=60x$$

$$32x^2+4x-1176=0$$

Using quadratic formula, we obtain $x=6$ (taking the positive result only).

Assuming that your result is correct, we are left with the quadratic equation $32x^2+4x-1179=0$. To solve for $x$, there are various techniques. The most common (and general) one would be completing the square, which will lead you to the general quadratic formula, allowing you to solve any quadratic equation of the form $ax^2+bx+c=0$ by
$$x=frac1{2a}left(-bpmsqrt{b^2-4ac}right).$$

Your equation is slightly incorrect. The actual equation is as follows(based on your question)

$$32x^2 +4x -1176=0$$

This equation can be factored using the quadratic formula

$$x=frac{-4pm sqrt{16+4cdot1176cdot32}}{2cdot32}$$

$$x=frac{-4pm sqrt{150544}}{64}$$

$$x=frac{-4pm 388}{64}$$

Since age can't be negative

$$x=(-4+388)/64 = 6$$

$32x^2 +4x -1176=0$ can be divided by $4$, resulting in $8x^2 + x - 294 = 0$.

The $ac$ method requires us to find two integers whose product is
$8 cdot (-294) = -2cdot2cdot 2 cdot 2 cdot 3 cdot 7 cdot 7$ and whose sum is $1$.

We note that $2$ cannot be a factor of both integers since then their sum will be a multiple of $2$ and $1$ is not a multiple of $2$. Likewise, 7 cannot be a factor of both integers. This leads us to guess that the two integers are

$$text{$-2cdot 2 cdot 2 cdot 2 cdot 3 = -48 qquad $ and $qquad 7 cdot 7 = 49$}$$

Next we replace $x =1x$ with $1x = -48x + 49x$ and factor.

begin{align}
32x^2 +4x -1176
&= 4(8x^2 + color{red}1x - 294) \
&= 4(8x^2 - color{red}{48}x + color{red}{49}x - 294) \
&= 4((8x^2 - 48x) + (49x - 294)) \
&= 4((8x(x - 6) + 49(x - 6)) \
&= 4(8x+49)(x - 6) \
end{align}