Solve for \(x\).

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\(\large \color{black}{\begin{align} (x^2+3x+1)(x^2+3x-3)\geq 5\hspace{.33em}\\~\\ \end{align}}\)
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One dumb method is to expand everything out and factor it again absorbing that 5 on right hand side

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And that gives $$ (x-1) (x+1) (x+2) (x+4)-5 $$ Then find for what values of x is this greater or equal to 0
Pretty sure you mean $$ (x-1) (x+1) (x+2) (x+4)\ge 0 $$ Then find for what values of x is this greater or equal to 0
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how did u factor that \(\large \color{black}{\begin{align} (x-1) (x+1) (x+2) (x+4)\ge 0\hspace{.33em}\\~\\ \end{align}}\)
x(x+3) +1 x(x+3) -3|dw:1436812802011:dw|
|dw:1436812913492:dw|
u know the rate of increase of both parabolas so u see the first instance they pass pr0duct 5
then from axis of symmetry at 1.5 u can see the other end too
@mathmath333 $$(x^2+3x+1)(x^2+3x-3)-5=(u+4)u-5=u^2+4u-5=(u+5)(u-1)$$for \(u=x^2+3x-3\)... and then we see that \((x^2+3x-3+5)(x^2+3x^2-3-1)=(x^2+3x+2)(x^2+3x-4)\) and both of these factor easily $$x^2+3x+2=(x+1)(x+2)\\x^2+3x-4=(x+4)(x-1)$$
so for \(f(x)=(x+1)(x+2)(x+4)(x-1)\ge 0\) we know that the solution is going to be every other interval between the roots, so either $$(-\infty,-4]\cup[-2,-1]\cup[1,\infty)\\\text{or}\\ [-4,-2]\cup[-1,1]$$
to determine which just test a point in one of the intervals: $$f(0)=(1)(2)(4)(-1)=-8<0$$so it follows the second intervals solve \(f(x)\le 0\) while the first solve \(f(x)\ge 0 \)
i d k why i am not able to see latex.
reload
reloaded twice.
well, you don't need to test both intervals; you already know that since each root has multiplicity 1 the function is crossing between each so alternating between positive and negative; then you just test a point in any one of the intervals to distinguish which of these two solutions is for nonnegative \(f\) and which is for nonpositive \(f\)
@oldrin.bataku Since I use table to determine it. It is like |dw:1436813611624:dw|
|dw:1436813832253:dw|
no I understand, I'm just saying that's more work than necessary, since you only need to check the sign of one of the intervals (not all of them) since we already determined the solution must either be \((-\infty,-4]\cup[-2,-1]\cup[1,\infty)\) or \([-4,-2]\cup[-1,1]\)
checking the sign on say \([-2,-1]\) is enough to tell you the sign on \((-\infty,-4]\cup[-2,-1]\cup[1,\infty)\) since the function is alternating in sign between intervals so it must have the same sign for all of these, and the opposite sign for \([-4,-2]\cup[-1,1]\)
Got you!!:)
@oldrin.bataku Is there any other way to solve it?
The best way is the @oldrin.bataku approach, if you can see it. Otherwise, (the hard way) is to us the Rational Root Theorem, which gives you the possible roots. After multiplying everything out we get: $$ (x^2+3x+1)(x^2+3x-3)-5=x^4+6 x^3+7 x^2-6 x-8 $$ The possible roots are then, \(\pm1,\pm2,\pm4,\pm8\). Then using synthetic division, test each. https://en.wikipedia.org/wiki/Rational_root_theorem https://en.wikipedia.org/wiki/Synthetic_division I knew a most there was 1 positive root, using Descartes' rule of signs, that also helped - https://en.wikipedia.org/wiki/Descartes%27_rule_of_signs I got lucky trying x=1,-1 first then everything else was easier because then I was just left with an easy quadratic to factor.
like the only other way I found is just a similar substitution but everything else is the same choose u=x^2+3x makes things factorable over the integers \[u=x^2+3x \\ (u+1)(u-3)-5 \ge 0 \\ (u^2-2u-3)-5 \ge 0 \\ u^2-2u-8 \ge 0 \\ (u-4)(u+2) \ge 0 \\ (x^2+3x-4)(x^2+3x+2) \ge 0 \\ (x-1)(x+4)(x+1)(x+2) \ge 0\]

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