And now, it's time for this week's synopsis critique! The author of COMPLEX SOLUTIONS, a YA Contemporary novel, submitted this synopsis. My in-line comments are [blue and in brackets], and I'll include a summary at the end. Feel free to comment below!
If you'd like a primer on how to write a synopsis, see my posts here and here. And if you want your synopsis critiqued on this website, fill out the form here, or email your 1-2 page synopsis to me at email@example.com. (NOTE: I'll email my critique to the author as soon as I'm done, so the author won't have to wait to see his/her synopsis on the site). Thanks for participating!
After the car accident that should have killed her too , sixteen-year-old LEXI THOMPSON lives with her grandmother. She wants to be accepted, to feel she’s not a freak.  Her response to cruel taunts and bullying from classmates is to cut herself, providing temporary relief from her emotional pain.
While she dreads the school scene, she loves math. She loves it the way her parents did.  The bonus is JOE BIONDI, a fellow math whiz, is in her AP Calculus class. He’s drawn to her shy, quiet nature, love of math and dedication to running. Running is the one thing—outside of cutting—that makes Lexi feel good.  Her relationship with Joe deepens, but her past secrets and continuing self-harm prevent her from getting too close. 
Lexi receives a letter from The Clay Mathematics Institute, a prestigious non-profit foundation, regarding a secret her parents kept while alive. Lexi discovers they likely proved the legendary Riemann Hypothesis, which she learns is a 150-year-old theorem regarding the behavior of prime numbers. The Clay Institute is offering one million dollars to the person who proves it.  With her parents gone, the whereabouts of their proof is unknown. Lexi is her parents  only hope of finding and submitting their proof, of saving their legacy and, in her mind, finally doing something that would make them proud.  It would also relieve some of the guilt she feels for surviving the accident. But she’s not the only one who knows their secret. DAVE EDISON, her father’s college buddy and later, his legal advisor, seeks the proof for his own glory. 
Lexi tells Joe about the lost proof and about Dave stalking her.  His  support helps her confess her self-harm and her past suicide attempt as well. Instead of pushing him away, Joe draws closer to her  and encourages her to seek help. She promises to do that once the research has been found and delivered to the Clay Institute.
Lexi and Joe finally find the proof on a USB stick, hidden in a photo album, but Dave Edison learns of their discovery and chases Lexi a few nights later on the beach. She outruns him, but afterward, is so rattled, she cuts herself too deep and Joe rushes her to get stitches at the local clinic. Without her knowledge, Joe gives the doctor at the clinic her grandmother’s number to call and consent to treatment. When Lexi is confronted by her grandmother afterward, she’s furious with Joe  and sneaks out the next morning to catch the bus to the Clay Institute to deliver the proof by herself.
Lexi runs into first, Joe, then Dave Edison  on the streets of Cambridge and ultimately, with the help of the local police, delivers the proof to the researchers at the Clay Institute  and Dave Edison is arrested.
All that remains is confirmation the theorem has been proven by Lexi’s parents and for Lexi to get the help for the underlying issues which cause her to self-harm. 
: Should have killed Lexi in addition to who? Who actually died?
: Why do people think she’s a freak? Because she was in a car crash (seems like she’d get sympathy for that, not bullying)? Or another reason?
: Did her parents die in the car crash?
: I thought math also made her feel good?
: This paragraph is a list of things Lexi likes. Work this into a few sentences that advance the plot. Describe how she and Joe grow closer. Describe how that impacts her character development. If running, is a part of that, then weave it in to the plot details.
: Did the Clay Institute know that Lexi’s parents proved the theorem? Why are they writing to Lexi now? Do they want her to bring them the proof? It sounds like they’re still looking for someone to solve it, so this confused me. Clarify this a bit.
: How would submitting her parents’ proof make them proud? It doesn’t seem like Lexi is actively doing anything here: she’s not solving the theorem herself. She’s just finding their proof and sending it in. I get the part about preserving their legacy, but it doesn’t seem like it would make them proud of her, because they’ve already done the work.
: If he’s not also a mathematician, how would he derive glory from the proof?
: When did Dave start stalking Lexi? Describe how this happens. Physically stalking her, stalking her online, etc.
: This is confusing as written. Reword to something like, ‘Instead of pushing him away, Lexi allows herself to grow closer to Joe, who encourages her to seek help.’
: Why would she be furious with Joe? It seems like he wouldn’t have had a choice in the matter, since Lexi’s a minor.
: Reword to something like, ‘On her way to the Clay Institute, Lexi runs into Joe’ and then describe what happens during that encounter. Then, pick up with ‘They see Dave…’ and describe what happens with him. Detailing what happens is the most important element of a synopsis.
: Make ‘Dave is arrested’ a separate sentence.
: This sentence is a bit confusing. Also, it makes it sounds like these things don’t actually happen in the book, and you’re setting them up for a sequel. If they do happen in the book, then say something like, ‘Lexi helps the Clay Institute confirm her parents’ proof by [describe how she does this].’ Then, ‘Finally, Lexi goes to her first therapy appointment to get help for the underlying issues causing her to self-harm.’ Also, I want a resolution to her relationship with Joe.
This is well-written, but it feels more like a long query than a synopsis. Take a step back and map out your main plot in terms of ‘who does what’ and ‘how does it advance the plot.’ Make sure you’re tracing all the significant events through the beginning, middle, and end of the story, and if you want to keep the synopsis at one page, delete the details that aren’t important (such as Lexi’s love of running). Also, make sure you keep your eye on your antagonist and track what he’s doing throughout the course of the story.
Best of luck with this!